## 3 Impact on System Performance: Simulations  
In this section we explore, by simulations, the impact of carrier induced frequency noise on system performance when a semiconductor laser is used in the transmitter module of an optical coherent link. This noise is mainly governed by the resonance frequency and the K-factor. We start by considering a laser with a lorentzian lineshape (flat spectral FM PSD). Figure 3 shows the simple schematic used in VPItransmissionMakerTM for this simulation. The transmitted signal is a 28 Gbaud dual polarization 16-level quadrature amplitude modulation (DP-16 QAM) carrying a pseudo-random bit sequence (PRBS) of length \(2^{15}-1\). In order ensure that only phase noise from the transmitter laser is present in the system, we model all of the components ideally, and set the standard single mode fiber (SSMF) length to 0. We use BPS algorithm for our CPR digital signal processing (DSP). We chose to implement BPS with 32 test phases, as it is found to be a good trade off between complexity and performance [21]. As expected, the penalty grows exponentially as the linewidth increases. We now proceed to include \(f_{R}\) and \(K\) parameters. Figure 4 shows the modified setup. The dual polarization IQ modulator (PolMux module) now receives an ideal laser phase-modulated with a time-domain noise sequence. We calculate that sequence using Matlab(r). To calculate that sequence, we first calculate the time-domain impulse response of the frequency model. This is obtained by the inverse Fourier transform of the square-root double sided FM PSD:  
\[h_{fn}=\mathcal{F}^{-1}\left\{\sqrt{S_{v}(f)}\right\}\,\ -F_{s}/2>f>F_{s}/2\]  
where \(F_{s}\) is the sampling frequency. The obtained filter \(h_{fn}\) is used to convolve an additive white gaussian noise (AWGN) sequence, which yields a time-domain frequency noise sequence \(fn=\mathrm{AWGN}*h_{fn}\). Since the frequency is the derivative of the phase, a cumulative sum of the obtained frequency noise sequence scaled by \(2\pi/F_{s}\) yields the \(L\) samples of the phase noise sequence:  
\[pn_{i}=\sum_{k=1}^{i}\frac{2\pi}{F_{s}}fn_{k}\,i=1,2,3...L\]  
which is used to phase-modulate the ideal laser, as illustrated in Figure 4. To evaluate the performance as a function of carrier induced frequency noise, we calculated 100 phase noise waveforms with \(K\)-factor on the range of 0.1 to 0.9 ns and resonance frequencies on the range of 1 to 10 GHz, while maintaining an intrinsic linewidth \(\Delta w_{int}=500\) kHz. The color plot shown in Figure 4 now substitutes the classical _Penalty vs. Linewidth_ (e.g., Figure 3), where the linewidth axis has been split into x and y, and the penalty is represented as a color. Penalties as high as 4.5 dB occur for the lowest values of \(K\) and \(F_{r}\). Note that, for the entire \(xy\) plane, the laser linewidth would still be calculated as 500 kHz. This is an important result because it proves that carrier induced frequency noise plays a role in system performance. To further investigate its implications, we proceed include different values of \(\Delta w_{int}\) as well: 50, 500 and 1000 kHz. We therefore need a volumetric color plot to represent the obtained penalties, as shown in Figure 5.  
Figure 3: Simulation setup for a lorentzian laser. Figure 5 shows the simulated penalty due to carrier induced frequency noise for different levels of intrinsic linewidth (\(\Delta v_{int}\)). Since there is no \(1/f\) noise, that would be the reported 3 dB linewidth of the laser. We continue to observe higher penalties as we approach lower values of \(K\) and \(F_{r}\) values. It becomes apparent that K factor plays a more important role than \(F_{r}\); however, both penalties are significantly diminished when \(F_{r}\) is higher than 1 GHz. We also observe that the evolution of carrier induced frequency noise penalty scales proportional to \(\Delta v_{int}\). For the case of \(F_{r}=1\) GHz and \(K=0.1\) ns, the recovered signal is above bit error rate (BER) of \(10^{-3}\) for values of \(\Delta v_{int}\) higher than 300 kHz.