## 2 Theory
This type of noise depends on the environmental and operating conditions, therefore the value of \(\kappa\) is merely for illustrative purposes. A plot of the full expression in (1) is given by the red curve, showing the excess FM-noise as well as the typical FM-noise of semiconductor lasers. The presence of the excess FM-noise term is typical of DBR-style lasers because the carrier density needed to tune the lasers remains unclamped, and hence carrier density noise transfers directly to the instantaneous frequency noise of the lasers [3; 18]. Now that we have given a brief overview of the tuning mechanism and FM-noise characterization, we will numerate the contribution of shot noise to excess FM-noise of DBR-style lasers. Figure 1: (**a**) Simplified schematic of a tunable distributed Bragg reflector (DBR) laser. (**b**) Schematic of the tuning mechanism of DBR lasers. In order to determine the FM-noise, we first need to determine the carrier dynamics. The rate equation describing carrier density dynamics in a passive section is described as:  
\[\frac{dN}{dt}=\frac{I}{eV}-R(N)+F_{N} \tag{2}\]  
where \(N\) is the carrier density, \(I\) is the injected current, \(e\) is quantum of electronic charge, \(V\) is the volume of the region that confines the carriers in the laser, \(R\) represents the carrier recombination rate, and \(F_{N}\) represents stochastic carrier recombination. \(R\) can be written as the sum of linear, bimolecular, and Auger recombination terms:  
\[R(N)=AN+BN^{2}+\mathcal{CN}^{3} \tag{3}\]  
The shot noise term is contained within \(I\), and we write \(I=I_{0}+\Delta i\), with \(I_{0}=\langle I\rangle\) representing the DC current, and \(\Delta i\) representing the shot noise. By definition for shot noise, the autocorrelation is:  
\[R_{\Delta i}(\tau)=\int\Delta i(t)\Delta i(t+\tau)dt=2 el_{0}\delta(\tau) \tag{4}\]  
and the corresponding power spectral density \(\Delta i\) is:  
\[S_{\Delta i}(f)=\int_{-\infty}^{\infty}R_{\Delta i}(\tau)\exp(-j2\pi f\tau)d \tau=2 el_{0} \tag{5}\]  
Figure 2: Typical FM-noise spectral density (SD) profile of tunable lasers. The blue curve represents a theoretical FM-noise SD curve of semiconductor lasers without any \(1/f\) noise nor excess FM-noise. The red curve shows the FM-noise of monolithic tunable lasers. The excess FM-noise portion is indicated, and this is the portion of the FM-noise curve that we concentrate on in this paper. At lower frequencies below 100 kHz, the \(1/f\) FM-noise dominates the FM-noise spectrum. The green line denotes the sole contribution to \(1/f\) FM-noise. Note that the \(1/f\) FM-Noise is added for illustrative purposes and that log scales were used for both axes. We approached the analysis by regarding that the system in (2) has reached a quasi-steady state with \(\left\langle N\right\rangle=N_{0}\) in response to the applied DC current \(I_{0}\). Therefore, \(N_{0}\) is the solution of:  
\[0=\frac{I_{0}}{eV}-R(N_{0}) \tag{6}\]  
Applying a small-signal analysis of (2), with \(N=N_{0}+\Delta n\), we obtain:  
Figure 2.0  
We define the carrier lifetime \(\tau_{N}\) as:  
Figure 2.1  
Plots of the steady state level of carrier density \(N_{0}\) and carrier lifetime \(\tau_{N}\) with respect to the bias current are shown in Figure 3 to reveal the sub-linear behavior of the static carrier density with respect to the bias current and the decrease in carrier lifetime with increasing bias current. The first and last terms on the right hand side of (7) describe the shot noise and random carrier recombination terms. The carrier recombination is considered to be a Markov process, with \(\left\langle F_{N}^{2}\right\rangle=2R(N_{0})/V\). The ratio of the magnitudes of the shot noise and dynamic carrier recombination terms is \(\left\langle\left(\Delta i/eV\right)^{2}\right\rangle/\left\langle F_{N}^{2}\right\rangle\), and was found to be equal in the magnitude of the Langevin term for the shot noise contribution \(\left\langle F_{N}^{2}\right\rangle=\left\langle\left(\Delta i/eV\right)^{2}\right\rangle\), using the relation in (6).