# Semiconductor Laser Linewidth Theory Revisited
## 2 Prerequisites and Basic Assumptions  
We consider edge-emitting semiconductor lasers as sketched in Figure 1 consisting of an arbitrary number of sections of different functionality. The transverse cross section is uniform within each section, but may vary from section to section. The preferred propagation direction of the optical field under lasing conditions is along the cavity axis parallel to the longitudinal coordinate \(z\). Thus laser can be well described by the traveling-wave ansatz for the main component of the electric field strength  
\[E(x,y,z,t)=\frac{e(x,y)}{\sqrt{2\varepsilon_{0}cn}}\Big{[}\Psi^{+}(z,t)\mathrm{ e}^{-i\tilde{P}_{0}z}+\Psi^{-}(z,t)\mathrm{e}^{i\tilde{P}_{0}z}\Big{]}\mathrm{e}^{i \omega_{0}t}+\mathrm{c.c.}, \tag{1}\]  
employing a scalar approximation and neglecting the dynamics in the transverse \((x,y)\)-plane. This means that we assume index-guiding in the transverse (lateral-vertical) plane and lasing in a single transverse mode and exclude gain-guiding, as appropriate for narrow-linewidth lasers. The real-valued field distribution \(e(x,y)\) is obtained as a solution of a real-valued waveguide equation normalized to \(\iint e^{2}(x,y)\,dxdy=1\)[29]. Moreover, \(\varepsilon_{0}\) is the vacuum permittivity, \(c\) the vacuum speed of light, \(n\) the reference modal index, \(\beta_{0}=2\pi n/\lambda_{0}\) the reference propagation constant, \(\lambda_{0}\) the reference wavelength and \(\omega_{0}=2\pi c/\lambda_{0}\) the reference frequency. The prefactor is chosen such that \(\|\Psi\|^{2}\equiv|\Psi^{+}|^{2}+|\Psi^{-}|^{2}\) is the optical power \(P\) with unit W. The left and right propagating fields \(\Psi^{\pm}\) vary slowly both in the longitudinal coordinate \(z\) and time \(t\) and fulfill the stochastic time-dependent coupled-wave equations [30; 31]  
\[\frac{n_{\mathrm{g}}(z)}{c}\frac{\partial}{\partial t}\Psi(z,t)+\bigg{[} \sigma_{z}\frac{\partial}{\partial z}+iM(z,N,\|\Psi\|^{2})+i\widehat{D}(z,t) \bigg{]}\Psi(z,t)=F_{\mathrm{sp}}(z,t,N) \tag{2}\]  
with  
\[\Psi(z,t)= \begin{bmatrix}\Psi^{+}(z,t)\\ \Psi^{-}(z,t)\end{bmatrix} ,\qquad\sigma_{z}= \begin{bmatrix}1&0\\ 0&-1\end{bmatrix} ,\]  
and  
\[M(z,N,\|\Psi\|^{2}) = \begin{bmatrix}\Delta\beta(z,N,\|\Psi\|^{2})&\kappa^{+}(z)\\ \kappa^{-}(z)&\Delta\beta(z,N,\|\Psi\|^{2})\end{bmatrix} , \tag{3a}\] \[\Delta\beta(z,N,\|\Psi\|^{2}) =\beta(z,N,\|\Psi\|^{2})-\beta_{0}=\frac{2\pi}{\lambda_{0}} \Delta n(z)+\beta_{N}(z,N,\|\Psi\|^{2}),\] (3b) \[\beta_{N}(z,N,\|\Psi\|^{2}) =\frac{2\pi}{\lambda_{0}}\Delta n_{N}(z,N,\|\Psi\|^{2})+\frac{i}{ 2}\Big{[}g(z,N,\|\Psi\|^{2})-\alpha(z,N,\|\Psi\|^{2})\Big{]}. \tag{3c}\]  
The Langevin forces  
\[F_{\mathrm{sp}}(z,t,N)= \begin{bmatrix}F_{\mathrm{sp}}^{+}(z,t,N)\\ F_{\mathrm{sp}}^{-}(z,t,N)\end{bmatrix} ,\]  
Figure 1: Schematic view of a two-section DBR laser consisting of an electrically biased gain section and a passive Bragg reflector section, exemplifying an edge-emitting multi-section semiconductor laser. model the spontaneous emission into the forward and backward traveling waves and are assumed to have zero mean  
\[\langle F_{\text{gp}}\rangle=0.\]  
Here, \(n_{\text{g}}\) is the real-valued group index, \(\beta\) is the propagation factor and \(\Delta n\) is the static index detuning with respect to the reference modal index. \(\beta_{N}\) describes the dependence of the propagation factor on the excess carrier density \(N\), consisting of a real part proportional to \(\Delta n_{N}\) and an imaginary part with \(g\) being the modal gain and \(\alpha\) the optical losses. \(\kappa^{\pm}\) are complex coupling coefficients describing a Bragg grating (coupling forward and backward traveling waves) and \(\widehat{D}(z,t)\) is the dispersion operator. The ensemble or temporal average is denoted by \(\langle\cdot\rangle\).