## 3 Results  
We create values for \(N\) using (2). First, we solved for \(N_{0}\) in (6) for a given value of bias current \(I_{0}\). Then, we created two uncorrelated zero-mean, unity-variance waveforms, with values taken from a Gaussian distribution to create \(F_{N}\) and the shot noise \((\Delta i)\). We took one of the random waveforms and scaled by \(\sqrt{2R(N_{0})/(Vt_{S})}\) to create the random carrier recombination, and created the shot noise \((\Delta i)\) term using the other random waveform and scaling by \(\sqrt{2I_{0}e/t_{S}}\), where \(t_{S}\) is the time step used to numerically solve (2). In all our simulations, \(t_{S}=10\) ps unless otherwise stated. Once the simulation reached quasi-steady state, i.e., when (6) was fulfilled, we gathered the waveform for \(N(t)\) and extracted the random part \(\Delta n(t)\). We calculated \(\Delta n(t)\) for ten different values of bias current applied to the tuning (phase) section, with the current values ranging from 1 to 10 mA. We calculated the spectral density of the carrier fluctuations by taking the squared magnitude of the resulting fast Fourier transforms (FFT) of \(\Delta n(t)\) and scaling by the spectral width of the frequency bins in the FFT.  
First, we calculated the SD of the carrier fluctuations and plotted each value of the current in Figure 4. The SD plots were averaged 1000 times for each value of injected current, and every simulation run used different uncorrelated numerical sources for \(F_{N}\) and \(\Delta i\). The calculated values for the SDs matched exactly with those given by the analytical formula for spectral density of the carrier noise in (10). Note that the roll-off in the FM-noise was due to the equivalent low-pass filtering effect due to the limited carrier dynamics [25; 26], with maximum frequency limited to 100 MHz. The horizontal lines in Figure 4 are just a guide to show that the calculated and analytical values match. We then converted the carrier fluctuations to a laser frequency variation using (11).  
The corresponding FM-noise SD plots are shown in Figure 5. For the parameters and operating conditions used in this paper, we found that the value for the FM-noise ranged from 2.3 MHz to 14.5 MHz, and these values were commensurate with experimental values for the FM-noise SD of this effect varying from 1 MHz to 15 MHz [15; 16; 18; 19], and around 10 MHz from the full simulation [20]. For completeness, we show the instantaneous frequency change of the laser over a 50 ns window in Figure 6. There are implications arising from the large instantaneous frequency shifts: For communication systems, these frequency shifts are directly translated to frequency offsets between the signal and local oscillator laser in the receiver. The frequency offsets add additional unwanted phase shifts from symbol to symbol, and these phase shifts need to be tracked so that they can be eliminated digitally in the receiver. A tuned second-order phase-locked loop [15] or use of modulation formats based on doubly differential phase shift keying alleviates the frequency offsets [17].  
From our simple analysis, we can deduce that shot noise from the bias currents is responsible for the excess FM-noise in tunable lasers. Once this had been verified, we were able to explore the implications of this finding by investigating the available options to reduce excess FM-noise in tunable lasers.  
Figure 4: Calculated SD of the carrier fluctuations. The bias current increases from 1 to 10 mA. The horizontal lines indicate analytical calculations of the carrier fluctuations SD. Note: log scales used.  
Figure 5: Excess FM-noise SD calculations corresponding to the carrier fluctuations in Figure 4. The bias current is increased from 1 to 10 mA. Note: log scales used.