# Semiconductor Laser Linewidth Theory Revisited
## 8 Impact of a Passive Section, Chirp Reduction Factor and the Fabry-Perot Case
### Linewidth of the Fabry-Perot Laser Cavity  
We start from (88) and employ the same approximations resulting in (100) and (101). The rate of spontaneous emission into the lasing mode (86) is  
\[R_{\rm sp}(N)=\frac{\int_{0}^{l}\|\Phi(z)\|^{2}n_{\rm sp}(N)g(N)\,dz}{\int_{0}^{ l}\|\Phi(z)\|^{2}\,{\rm Re}\Big{(}\frac{\partial\bar{\theta}}{\partial\omega} \Big{)}\,dz}=\frac{c}{n_{\rm g}}n_{\rm sp}(N)g(N). \tag{105}\]  
The expression (105) should be compared with  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]  
often used in rate equation based modeling [56], where \(\beta_{\rm sp}\) is the dimensionless spontaneous emission factor (ratio between the spontaneous emission going into the lasing mode and the spontaneous emission into all modes), \(V\) the volume of the active region and \(R_{\rm rad}=BN^{2}\) the rate of radiative spontaneous recombination (\(B\) bimolecular recombination coefficient). Equalizing (105) and (106) at the lasing threshold yields  
\[\beta_{\rm sp}=\frac{c}{n_{\rm g}}\frac{n_{\rm sp}(N_{\rm th})g(N_{\rm th})}{ VR_{\rm rad}(N_{\rm th})}, \tag{107}\]  
which is of the order \(10^{-5}\) for typical edge-emitting lasers. The photon number (83) can be written as  
\[I_{\rm ph} =\frac{P_{0}}{\hbar\omega_{0}}\frac{\int_{0}^{l}\|\Phi(z)\|^{2} \,{\rm Re}\Big{(}\frac{\partial\bar{\theta}}{\partial\omega}\Big{)}\,dz}{(1-| r_{0}|^{2})|\Phi^{-}(0)|^{2}} \tag{108}\] \[=\frac{n_{\rm g}}{c}\frac{P_{0}}{\hbar\omega_{0}((1-|r_{0}|^{2}) |\[PERSON]^{-}(0)|^{2})}\int_{0}^{l}\Big{[}|\[PERSON]^{+}(0)|^{2}{\rm e}^{2\,{\rm Im}( \overline{\beta})z}+|\[PERSON]^{-}(0)|^{2}{\rm e}^{-2\,{\rm Im}(\overline{\beta})z }\Big{]}\,dz\] \[=\frac{n_{\rm g}}{c}\frac{P_{0}}{\hbar\omega_{0}}\frac{|r_{0}|^{2 }\Big{(}{\rm e}^{2\,{\rm Im}(\overline{\beta})l}-1\Big{)}-{\rm e}^{-2\,{\rm Im }(\overline{\beta})l}+1}{2\,{\rm Im}(\overline{\beta})(1-|r_{0}|^{2})}.\]  
The complex mode frequencies of the FP cavity (obtained from (89) applying the boundary conditions (12) at \(z=0\) and \(z=l\)) are the solutions of  
\[\Omega_{m}=\frac{c}{n_{\rm g}}\Big{[}\frac{m\pi}{l}-\overline{\beta}-\frac{i} {2l}\ln\big{(}r_{0}r^{+}(\Omega_{m})\big{)}\Big{]} \tag{109}\]  
with \(m\in\mathbb{Z}\). To ensure \({\rm Im}(\Omega)=0\) for the lasing mode, (109) leads to the usual threshold condition  
\[\alpha_{\rm out}\equiv-\frac{1}{l}\ln(|r_{0}r^{+}({\rm Re}(\Omega))|)=2\,{\rm Im }(\overline{\beta})\equiv g-\alpha. \tag{110}\]  
In what follows, we set \(r_{l}=r^{+}(\Omega)\) (reflectivity seen by the laser at \(z=l\)). From (110) it follows  
\[I_{\rm ph}=\frac{n_{\rm g}}{c}\frac{P_{0}}{\hbar\omega_{0}}\frac{|r_{0}|^{2} \Big{(}\frac{1}{|r_{0}r_{l}|}-1\Big{)}-|r_{0}r_{l}|+1}{2\,{\rm Im}(\overline{ \beta})(1-|r_{0}|^{2})}=\frac{n_{\rm g}}{c}\frac{P_{\rm out}}{\hbar\omega_{0} \omega_{\rm out}}, \tag{111}\]  
where we used the relation between outcoupled power \(P_{0}\) and total output power \(P_{\rm out}\)  
\[P_{0}=\frac{P_{\rm out}}{1+\frac{1-|r_{l}|^{2}}{1-|r_{0}|^{2}}\frac{|r_{0}|}{ |r_{l}|}}. \tag{112}\]  
The final result is  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]0where Petermann's \(K\)-factor  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]1  
of the FP cavity can be similarly derived as  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]2  
in agreement with [9; 27]. Equation (113) together with (112) was used by several authors, e.g., [12; 55; 57]. However, one should be aware of the approximations involved in the derivation. The original Schawlow-Townes formula [1] is obtained by setting \(K_{\text{FP}}=1\), \(n_{\text{sp}}=1\), \(\alpha=0\), \(\alpha_{\text{H}}=0\), and relating the outcoupling losses \(\alpha_{\text{out}}\) to the spectral FWHM of a cavity resonance by \(\Delta v_{\text{cav}}=c/n_{\text{g}}\cdot\alpha_{\text{out}}/2\pi\) to  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]3  
There are two reasons why this differs from the original formula by a factor of 4. First, in [1] the half widths instead of full widths are used (factor of 2).