## 2 Theory  
We will first briefly review and define the tuning mechanism of tunable DBR-style lasers. More comprehensive explanations can be found in [3; 4; 5; 6]. We concentrate on lasers with a single DBR grating in this paper, though the analysis contained within this paper applies equally to other tunable lasers with sampled DBR gratings that provide for an extended wavelength tuning range exceeding 40 nm [3]. A typical DBR laser structure is shown in Figure 1a. Independent amounts of current are pumped into the different sections. The gain section provides the optical gain and thus invokes the overall lasing action. The current in the other two sections only varies the refractive index (and loss) of those sections, that allow for the laser to be tuned. Figure 1b depicts the tuning mechanism. There are many allowable longitudinal modes determined by the effective length of the device, and the DBR grating section selects a single lasing longitudinal mode, and the phase (tuning) section provides for small tuning of the laser wavelength by adjusting the overall optical length of the device. Preliminary to the FM-noise analysis, we provide a brief overview of the FM-noise of semiconductor lasers. There is an equivalence between laser linewidth and FM-noise SD, as explained thoroughly in [24]. A purely Lorentzian lineshape corresponds to a flat FM-noise spectral density, whose constant value is proportional to the '3 dB linewidth' [24]. To convert between FM-noise SD and '3 dB linewidth', one needs to multiply the FM-noise by \(2\pi\)[24]. Due to the common usage of '3 dB linewidth' throughout the photonics community, we chose to scale all of our calculations for FM-noise SD by \(2\pi\). Unlessstated otherwise, all of the values for the FM-noise SD in this paper are scaled by \(2\pi\). Concentrating on evaluating the FM-noise SD allows us to clearly distinguish between the various independent contributions to the laser phase noise. A general expression of the FM-noise spectral density of these DBR lasers is given by [18]:  
\[|\Delta\widetilde{v}_{FM}(f)|^{2}=\frac{|\Delta\widetilde{v}_{STH}|^{2}}{1+ \alpha_{H}^{2}}\left[1+\frac{\alpha_{H}^{2}f_{r}^{4}}{\left(f_{r}^{2}-f^{2} \right)^{2}+\left(\frac{\Gamma f}{2\pi}\right)^{2}}\right]+\frac{|\Delta \widetilde{v}_{L}|^{2}}{1+\left(\frac{f}{f_{c}}\right)^{2}}+\frac{\kappa}{f} \tag{1}\]  
We plotted an exemplar curve using this expression in Figure 2. The first term on the right hand side of (1) is the usual '3 dB linewidth', combined with the relaxation of semiconductor lasers as derived by [PERSON] in [23]. A plot of this first term is given by the blue curve in Figure 2. As the name implies, FM-noise SD is analogous to power spectral density, hence the appearance of the terms \(|.|^{2}\) in (1). Therefore, the units for FM-noise spectral density conveniently happen to be Hz, though to maintain consistency throughout this paper where we will be analytically solving stochastic differential equations with [PERSON] contributions, we refer to these SD quantities as \(|.|^{2}\). \(|\Delta\widetilde{v}_{STH}|^{2}\) is the FM-noise spectral density that would be interpreted as the \"Schaowlow-Townes-Henry\" or \"3 dB linewidth\" [23], \(f_{r}\) is the relaxation oscillation frequency, and \(\Gamma\) is the damping rate of the relaxation oscillation. Typically, semiconductor lasers are under-damped, hence the peak in the FM-noise SD plot. \(\alpha_{H}\) is [PERSON]'s linewidth enhancement factor. The second term on the RHS of (1) describes the excess FM-noise or filtered FM-noise, and has a low-pass type response with cut-off frequency corresponding to the modulation dynamics of the passive tuning section [18; 25; 26]. It is this type of FM-noise that we are studying in this paper. The third term represents the typical \(1/f\) noise, which is prevalent at low frequencies and is caused by \(1/f\) noise on the drive currents and environmental factors [21; 22].