## 2 Semiconductor Laser Frequency Noise Model  
The short-term phase variation of a waveform, can be represented either as phase or frequency noise. Frequency noise refers to random fluctuations of the instantaneous frequency, which is the temporal derivative of the phase. We model the single-sided PSD of frequency noise per frequency bandwidth as:  
\[S_{v}(f)=\frac{10^{9}\Delta v_{(1/f)}}{\pi f}+\frac{\Delta v_{int}}{\pi(1+ \alpha^{2})}\left(1+\alpha^{2}\frac{f_{R}^{4}}{(f_{R}^{2}-f^{2})^{2}+\left( \frac{kf_{R}^{2}}{2\pi}f\right)^{2}}\right) \tag{1}\]  
where:  
* \(\Delta v_{(1/f)}\) describes the level of \(1/f\) noise at 1 GHz. * \(\Delta v_{int}\) describes the level of the intrinsic frequency noise at low frequencies. * \(f_{R}\) is the resonance frequency. * \(K\)-factor describes how the damping rate increases with relaxation frequency. The \(K\)-factor of semiconductor lasers is approximately bias independent and in the range of 0.1-1 ns. * \(\alpha\) parameter determines the magnitude of the carrier induced noise. Figure 1 shows the frequency PSD and optical lineshape for a frequency modulation (FM) particular noise model. It is a common practice to scale the result by \(\pi\) in order to match the intrinsic linewidth (\(\Delta v_{int}\)) with what would be the 3 dB linewidth if the lineshape was Lorentzian (white frequency noise) [16]. The phase noise PSD can be obtained by simply dividing the frequency noise PSD by \(f^{2}\). If the frequency noise PSD is known, the optical spectrum can be calculated as the Fourier transform of the autocorrelation function [5]:  
\[S_{v}(\Delta v)=\mathscr{F}[R(\tau)]\propto\] \[\mathscr{F}\left\{\exp\left[-2(\pi\tau)^{2}\int_{0}^{\infty}S_{v} (f)\left|\frac{\sin\left(\pi f\tau\right)}{\pi f\tau}\right|^{2}df\right]\right\} \tag{2}\]where \(S_{v}(\Delta v)\) is the optical spectral density, \(\Delta v=v-v_{0}\) is the optical frequency measured with laser frequency \(v_{0}\) as a reference, and \(S_{v}(f)\) is the frequency noise PSD. Figure 2 shows how the 3 dB linewidth of a laser changes as a function of each of the five parameters of the model. Each parameter is swept while leaving all others fixed. On the five upper plots in Figure 2, the red curve shows the FM PSD (\(\pi S_{v}(f)\)) at the start of the sweep, while the blue curve shows it at the end point of the sweep. The lower plots show the measured 3 dB linewidth as a function of each parameter under the sweep range. For \(\Delta v_{int}\), we can observe a linear relation with respect the \(S_{v}(\Delta v)\). This is due to the fact that this parameter scales the amount of white frequency noise evenly throughout the FM noise spectrum. In case of \(\Delta v_{(1/f)}\), the 3 dB linewidth scales with the \(\sqrt{\Delta v_{(1/f)}}\). As for \(f_{R}\), \(K\) and \(\alpha\) parameters, there is no impact on laser linewidth for the values under study. \(1/f\) frequency noise has a large impact on laser linewidth, but low impact on system performance when exhibited by the transmitter laser [17; 18; 19; 20]. However, the implications of \(f_{R}\), \(K\) and \(\alpha\) of a semiconductor laser having no impact on its linewidth are major when considering how the optical communications community benchmarks CPR algorithms. This is usually reported as a penalty versus the laser linewidth times the symbol period. This has worked well for ECL lasers where the FM PSD is approximately flat up to the receiver bandwidth, in which case we obtain a lorenztian lineshape where the 3 dB linewidth scales linearly with the level of white frequency noise (as shown in the first sweep of Figure 2). However, when taking into account the carrier induced frequency noise of semiconductor lasers (\(f_{R}\) and \(K\)) this figure of merit no longer represents the amount of frequency noise, and hence phase noise in the system. Therefore, this paper is focused only on carrier induced frequency noise.