# Semiconductor Laser Linewidth Theory Revisited
## 2 Prerequisites and Basic Assumptions
The dependence of \(\Delta n_{N}\), \(g\) and \(\alpha\) on the optical power \(\|\Psi\|^{2}\) is due to nonlinear effects like the Kerr effect, gain compression, and two-photon absorption, which affect the stationary states only slightly but can have a significant impact on the dynamic properties. All quantities entering (2) are obtained by weighting the original quantities entering Maxwell's equations with the intensity distribution \(e^{2}(x,y)\) of the transverse mode. If we choose the origin of \(z\) such that the grating given by a complex dielectric function \(\varepsilon(x,y,z)\) has the property \(\varepsilon(x,y,z)=\varepsilon(x,y,-z)\), then \(\kappa^{+}=\kappa^{-}=\kappa\) holds. Equation (2) has to be solved subject to the usual boundary conditions  
\[\Psi^{+}(0,t)=r_{0}\Psi^{-}(0,t)\quad\text{and}\quad\Psi^{-}(L,t)=r_{L}\text{e} ^{-2i\beta_{0}L}\Psi^{+}(L,t), \tag{4}\]  
with \(L\) being the total cavity length and \(r_{0}\) and \(r_{L}\) the complex-valued reflection coefficients at the facets at \(z=0\) and \(z=L\), respectively. Scattering matrices establish transition conditions on \(\Psi\) at the interfaces between different sections [22; 30]. The coupled-wave Equation (2) must be supplemented with an equation for the (excess) carrier density \(N\). Although the theoretical treatment is quite general, we will specifically treat diode lasers in this paper, where the carrier dynamics is governed actually by both electrons and holes. We assume charge neutrality and consider only the excess carriers in the active region with density \(N=n-n_{0}=p-p_{0}\) (where \(n\) and \(p\) are the electron and hole densities and \(n_{0}\), \(p_{0}\) are the corresponding equilibrium densities), neglecting any transport and capture effects. Thus, the excess carrier density is assumed to obey the rate equation  
\[\frac{\partial N}{\partial t}-\frac{j(z,N)}{qd}+R(z,N)+R_{\text{st}}(z,N,\| \Psi\|^{2})=F_{N}(z,t,N) \tag{5}\]  
with the rate of spontaneous and non-radiative recombination \(R\) and the rate of stimulated recombination  
\[R_{\text{st}}=\frac{\text{Re}\Big{(}\Psi^{*}\cdot[g-2i\widehat{D}]\Psi\Big{)} }{dW\hbar\omega_{0}} \tag{6}\]  
which can be derived by multiplying (2) from the left with \([\Psi^{+*},\Psi^{-*}]\), the complex conjugate of (2) with \([\Psi^{+},\Psi^{-}]\) and adding both equations to obtain a balance equation for the electromagnetic energy in differential form ([PERSON]'s theorem) [30]. Here the dot product means \(\Psi_{1}\cdot\Psi_{2}=\Psi_{1}^{+}\Psi_{2}^{+}+\Psi_{1}^{-}\Psi_{2}^{-}\) and \(q\) is the elementary charge, \(\hbar\) the reduced [PERSON]'s constant and \(d\) and \(W\) are the thickness and width, respectively, of the active region. The current density can be related to the (sectional) applied voltage by means of the [PERSON] model [32; 33]  
\[j(z,N)=\frac{U_{s}-U_{\text{F}}(N)}{WL_{s}R_{s}} \tag{7}\]  
with sectional applied voltage \(U_{s}\), resistance \(R_{s}\), injection current \(I_{s}\), length \(L_{s}\), and the Fermi voltage \(U_{\text{F}}\). The carrier-density dependent injection current density given by (7) counteracts longitudinal spatial hole burning, which substantially lowers the degradation of the side mode suppression in strongly coupled DFB lasers compared to a model assuming a constant current density [34]. The frequency modulation response, albeit not topic of the present paper, is also affected as shown in [35]. Dispersion, i.e., the frequency dependence of the dielectric function, enters (2) first via the group index \(n_{\text{g}}\) in front of the time derivative resulting from a linearization of the real part of the dielectric function with respect to the frequency \(\omega\). Second, the dispersion of the optical gain (and associated refractive index) is described in the time domain by the operator \(\widehat{\mathcal{D}}(z,t)\), which can be obtained, e.g., by approximating the gain spectrum by a Lorentzian (or a series of Lorentzians) and a back-transformation into the time domain.