## 2 Theory
Assuming that the current noise and carrier recombination have zero correlation, we may add their power spectral densities. Taking the Fourier transform of (7) and calculating the power spectral density, we obtain:  
Figure 2.2  
At low frequencies for which \(f<<\left(2\pi\tau_{N}\right)^{-1}\), the power spectral density of the carrier fluctuation is:  
Figure 2.3  
Figure 2.4 **(a)** Calculated static carrier density with respect to injected current. (**b**) Calculated carrier lifetime with respect to injected current. The values of the laser parameters are listed in Figure 2.5  
In order to relate carrier density changes to instantaneous changes in the lasing frequency, we write [3]:  
Figure 2.6  
with \(v_{L}\) denoting the lasing frequency and \(n_{T}\) denoting the refractive index of the tuning section. We have already described \(\Delta n\) due to the shot noise. To estimate for \(dv_{T}/dn_{T}\), we can approximate the lasing frequency if we know the longitudinal mode spacing and longitudinal mode number. Ignoring dispersion in the frequency dependence of the refractive index, the lasing frequency is an integer multiple of the free spectral range (FSR), \(v_{L}=m\Delta v_{FSR}\). \(\Delta v_{FSR}\) is related to the length and refractive index of the laser, and invoking an approximation in [5], we write:  
Figure 2.7  
where \(n_{eq}\) and \(L_{eq}\) are the equivalent refractive index and length of the laser gain and DBR sections. For the tuning section (subscripted by \(T\)), \(n_{T}\) is the refractive index and \(L_{T}\) is the length of the tuning section. Using the parameters given in Table 1, we estimate that \(\Delta v_{FSR}\) = 42 GHz. Given a lasing frequency of ~193 THz, we estimate \(m\) to be of the order of 4096. We can now estimate for \(dv_{L}/dn_{T}\) using (12) as:  
Figure 2.8  
To calculate \(dn_{T}/dN\), we first invoke the Drude model for the refractive index of a free-carrier plasma [3]:  
Figure 2.9  
with \(m_{e}^{*}\) being the equivalent effective mass of electrons and holes in the InGaAsP region and \(\varepsilon_{0}\) being the permittivity of free space. In addition to the free carrier plasma, the contribution due to band filling increases the change in refractive index by a factor of \(K\) = 2 [3]. Therefore, assuming that the group refractive index equals the refractive index, we can write:  
Figure 2.0 \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Quantity** & **Value and Units** \\ \hline \(A\) & Shockley–Read–Hall recombination coefficient & \(8\times 10^{8}\) s\({}^{-1}\) \\ \(B\) & Bimolecular recombination coefficient & \(1\times 10^{-17}\) m\({}^{3}\)/s \\ \(C\) & Auger recombination coefficient & \(1\times 10^{-41}\) m\({}^{6}\)/s \\ \(e\) & Quantum of electronic charge & \(1.6\times 10^{-19}\) C \\ \(V\) & Volume of tuning section & \(10\times 10^{-18}\) m\({}^{3}\) \\ \(n_{\text{eq}}\) & Equivalent refractive index of laser other than the tuning section & 3.5 \\ \(L_{\text{eq}}\) & Equivalent length of laser excluding the tuning section & 1 mm \\ \(n_{\text{T}}\) & Refractive index of tuning section & 3.5 \\ \(\kappa\) & Strength of 1/f FM noise & \(1\times 10^{10}\) s \\ \(L_{\text{T}}\) & Length of tuning section & 100 μm \\ \(v_{L_{0}}\) & Lasing wavelength with \(n_{\text{T}}\) = 3.5 & 193.00 THz \\ \(m_{e}^{*}\) & Equivalent effective mass of electrons in InGaAsP & 0.02 \(\times\) 9.1 \(\times\) 10\({}^{-31}\) kg \\ \(\varepsilon_{0}\) & Permittivity of free space & 8.85 \(\times\) 10\({}^{-12}\) F/m \\ \(K\) & Excess refractive index factor & **4** \\ \hline \end{tabular}
\end{table} 
Table 1: Laser parameters and physical constants. For this paper, we set \(K=1\). Nonetheless, this factor is material-dependent, and setting \(K=1\) allows us to set an easily scalable benchmark for the FM-noise SD arising from shot noise.