# Semiconductor Laser Linewidth Theory Revisited
## 5 Correlation Functions  
The correlation function of the spontaneous emission noise \(F_{f}\) defined in (23) is  
\[\langle F_{f}^{*}(t)F_{f}(t^{\prime})\rangle=\frac{\langle(\Phi^{*},F_{\text{ sp}}^{*}(z,t))(\Phi,F_{\text{sp}}(z^{\prime},t^{\prime}))\rangle}{\Bigl{|} \bigl{(}\Phi,\frac{\hat{\omega}\hat{\rho}}{\hat{\omega}\hat{\omega}}\Phi\bigr{)} \Bigr{|}^{2}}. \tag{54}\]  
The numerator reads  
\[ \begin{split}&\langle(\[PERSON]^{*},F_{\text{sp}}^{*}(z,t))(\Phi,F_{ \text{sp}}(z^{\prime},t^{\prime}))\rangle\\ &=\int_{0}^{L}\int_{0}^{L}\Bigl{\langle}\Bigl{[}\Phi^{++}F_{ \text{sp}}^{-*}(z,t)+\Phi^{-*}F_{\text{sp}}^{++}(z,t)\Bigr{]}\Bigl{[}\[PERSON]^{+ }F_{\text{sp}}^{-}(z^{\prime},t^{\prime})+\[PERSON]^{-}F_{\text{sp}}^{+}(z^{\prime },t^{\prime})\Bigr{]}\Bigr{\rangle}\,dzdz^{\prime}\\ &=\int_{0}^{L}\int_{0}^{L}\Bigl{[}|\[PERSON]^{+}|^{2}\langle F_{\text{ sp}}^{-*}(z,t)F_{\text{sp}}^{-}(z^{\prime},t^{\prime})\rangle+|\[PERSON]^{-}|^{2} \langle F_{\text{sp}}^{+*}(z,t)F_{\text{sp}}^{+}(z^{\prime},t^{\prime}) \rangle\Bigr{]}\,dzdz^{\prime}\\ &=2\int_{0}^{L}\|\[PERSON](z)\|^{2}D_{\text{sp}}(z)\,dz\,\delta(t-t^{ \prime})\end{split}  \tag{55}\]  
with  
\[\|\[PERSON]\|^{2}=|\[PERSON]^{+}|^{2}+|\[PERSON]^{-}|^{2} \tag{56}\]  
taking into account the noise covariance functions [25; 26; 27] for index and absorption coupling  
\[ \begin{split}&\langle F_{\text{sp}}^{\pm*}(z,t)F_{\text{sp}}^{ \pm}(z^{\prime},t^{\prime})\rangle=2D_{\text{sp}}(z)\delta(t-t^{\prime}) \delta(z-z^{\prime}),\\ &\langle F_{\text{sp}}^{\mp*}(z,t)F_{\text{sp}}^{\pm}(z^{\prime}, t^{\prime})\rangle=0,\\ &\langle F_{\text{sp}}^{\pm}(z,t)F_{\text{sp}}^{\pm}(z^{\prime}, t^{\prime})\rangle=0\\ &\langle F_{\text{sp}}^{\mp}(z,t)F_{\text{sp}}^{\pm}(z^{\prime}, t^{\prime})\rangle=0\end{split}  \tag{57}\]with  
\[2D_{\text{sp}}(z,N,\|\Psi\|^{2})=\hbar\omega_{0}n_{\text{sp}}(N)g(z,N,\|\Psi\|^{2}), \tag{58}\]  
where \(n_{\text{sp}}\) is the population inversion factor (a Bose-Einstein distribution function multiplied by \(-1\)). The correlation functions for gain-coupling can be found in [47]. Since \(\Psi\) has the unit \(\sqrt{\text{W}}\), \(D_{\text{sp}}\) must have the unit \(\text{Ws}/\text{m}\) as it is. The diffusion coefficient \(D_{f^{*}f}\) defined by  
\[\langle F_{f}^{*}(t)F_{f}(t^{\prime})\rangle=2D_{f^{*}f}\delta(t-t^{\prime}). \tag{59}\]  
is therefore  
\[D_{f^{*}f}=D_{f^{*}f}(N,\|\Psi\|^{2})=\frac{\int_{0}^{L}\|\Phi(z)\|^{2}D_{\text {sp}}(z,N,\|\Psi\|^{2})\,dz}{\left|(\Phi,\frac{\partial\rho}{\partial\omega} \Phi)\right|^{2}}. \tag{60}\]  
The correlation functions of the Langevin forces entering (37) and (38) read  
\[\langle F_{|f|^{2}}(t)F_{|f|^{2}}(t^{\prime})\rangle =2D_{|f|^{2}|f|^{2}}\delta(t-t^{\prime})=4D_{f^{*}f}\langle|f|^{2 }\rangle\delta(t-t^{\prime}), \tag{61}\] \[\langle F_{\varphi}(t)F_{\varphi}(t^{\prime})\rangle =2D_{\varphi\varphi}\delta(t-t^{\prime})=\frac{D_{f^{*}f}}{\langle |f|^{2}}\delta(t-t^{\prime}),\] (62) \[\langle F_{|f|^{2}}(t)F_{\varphi}(t^{\prime})\rangle =2D_{|f|^{2}\varphi}\delta(t-t^{\prime})=0, \tag{63}\]  
which can be derived using the transformation rules for the diffusion coefficients [23]  
\[D_{|f|^{2}|f|^{2}} =\Big{(}\frac{\partial|f|^{2}}{\partial f}\frac{\partial|f|^{2}} {\partial f^{*}}+\frac{\partial|f|^{2}}{\partial f^{*}}\frac{\partial|f|^{2}} {\partial f}\Big{)}D_{f^{*}f},\] \[D_{\varphi\varphi} =\Big{(}\frac{\partial\varphi}{\partial f}\frac{\partial\varphi} {\partial f^{*}}+\frac{\partial\varphi}{\partial f^{*}}\frac{\partial\varphi} {\partial f}\Big{)}D_{f^{*}f}, \tag{64}\] \[D_{|f|^{2}\varphi} =\Big{(}\frac{\partial|f|^{2}}{\partial f}\frac{\partial\varphi} {\partial f^{*}}+\frac{\partial|f|^{2}}{\partial f^{*}}\frac{\partial\varphi} {\partial f}\Big{)}D_{f^{*}f},\]  
and the relations \(|f|^{2}=f^{*}f\) and \(\varphi=-\frac{i}{2}\text{ln}(f/f^{*})\).