# Semiconductor Laser Linewidth Theory Revisited
## 3 Equation of the Field Amplitude  
Throughout this work, we restrict ourselves to the analysis of the noise at continuous wave (CW) emission, i.e., at steady-state lasing. Therefore, we expand \(\Psi(z,t)\) in terms of the stationary eigenmodes \(\Phi_{m}(z)\) (longitudinal modes),  
\[\Psi(z,t)=\sum_{m}f_{m}(t)\Phi_{m}(z), \tag{8}\]  
satisfying the equation  
\[\bigg{[}\sigma_{z}\frac{\partial}{\partial z}+iM\Big{(}z,\overline{N},\overline {\|\Psi\|^{2}}\Big{)}+i\beta_{\mathcal{D}}(z,\Omega_{m})+i\frac{n_{\text{g} }(z)}{c}\Omega_{m}\bigg{]}\Phi_{m}(z)=0 \tag{9}\]  
with  
\[M\Big{(}z,\overline{N},\overline{\|\Psi\|^{2}}\Big{)} = \begin{bmatrix}\Delta\beta\Big{(}z,\overline{N},\overline{\| \Psi\|^{2}}\Big{)}&\kappa(z)\\ \kappa(z)&\Delta\beta\Big{(}z,\overline{N},\overline{\|\Psi\|^{2}}\Big{)} \end{bmatrix} \equiv\overline{M}, \tag{10}\] \[\Delta\beta\Big{(}z,\overline{N},\overline{\|\Psi\|^{2}}\Big{)} =\beta\Big{(}z,\overline{N},\overline{\|\Psi\|^{2}}\Big{)}-\beta _{0}\equiv\overline{\beta}-\beta_{0}=\Delta\overline{\beta}. \tag{11}\]Here, \(\overline{N}\) and \(\overline{\|\Psi\|^{2}}\) are the steady-state carrier density and optical power, respectively, at which the operator of the eigenvalue problem (9) is expanded (\(\overline{\beta}\), \(\overline{M}\) are the corresponding propagation factor and matrix \(M\)). The complex eigenvalue \(\Omega_{m}\) describes both, the frequency deviation \(\text{Re}(\Omega_{m})=\omega_{m}-\omega_{0}\) (or the wavelength deviation \(\Delta\lambda_{m}=\text{Re}(\Omega_{m})\left.d\lambda/d\omega\right|_{\lambda_ {0}}\)), and the mode damping rate \(\text{Im}(\Omega_{m})\). The dispersion \(\beta_{\mathcal{D}}(z,\Omega_{m})\) is given by the operator \(\mathcal{D}(z,t)\) in the frequency domain. Equation (9) has to be solved subject to the boundary conditions  
\[\Phi_{m}^{+}(0) =r_{0}\Phi_{m}^{-}(0) \tag{12}\] \[\Phi_{m}^{-}(L) =r_{L}\text{e}^{-2i\beta_{0}L}\Phi_{m}^{+}(L)\]  
following from (4). In the case of stationary eigenmodes considered here, the two-point boundary value problem (9) together with the boundary conditions (12) is in fact a quasi-linear eigenvalue problem. The point of expansion \((\overline{\|\Psi\|^{2}},\overline{N})\) of the nonlinear operator must be chosen self-consistently, i.e., it is required to satisfy the conditions (36) and (68) for the (mean) stationary state given below. Furthermore, in the presence of dispersion \(\beta_{\mathcal{D}}(\Omega_{m})\), the eigenvalue problem is also nonlinear in the eigenvalue. The problem is similar to the self-consistent field method in electronic structure theory [43] and can be treated in a similar way (e.g., linearization of the optical-power and the frequency dependency of the gain dispersion and fixed-point iteration, where the point of expansion is updated to the most recent stationary state in each step). Alternatively, the system can be directly propagated in the time-domain (as it has been done in Section 10), until convergence to a suitable CW state has been achieved. Note that different modes (eigensolutions of (9)) fulfill the orthogonality relation  
\[(\Phi_{m},\frac{\partial\beta}{\partial\omega}\Phi_{n})=0\quad\text{for} \quad m\
eq n \tag{13}\]  
which can be derived by multiplying (9) from the left by \([\Phi_{n}^{-},\Phi_{n}^{+}]\), the corresponding equation for \(\Phi_{n}\) by \([\Phi_{m}^{-},\Phi_{m}^{+}]\), integrating along \(z\), using (12) and subtracting both equations. Here the short notation  
\[(\Phi,\Psi)=\int_{0}^{L}(\Phi^{+}\Psi^{-}+\Phi^{-}\Psi^{+})\,dz \tag{14}\]  
denotes an inner product, distinguished from a standard Hilbert space scalar product. At an exceptional point the expansion (8) includes also a generalized eigenmode [10].