## 4 Implications  
The simple analysis in the previous section allows us to explore how to minimise the FM-noise in tunable lasers. The primary goal of a tunable laser is to be able to tune the lasing frequency of the laser by one whole FSR. Therefore, our figure of merit for each laser is to establish the current needed to tune that laser by one whole FSR. The carrier density required to tune the laser by one FSR using (11) and (12) is:  
\[\Delta v_{FSR}=\frac{c}{2\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}}=\bigg{|}\frac{ dv}{dn}\bigg{|}\bigg{|}\frac{dn}{dN}\bigg{|}\Delta N_{FSR} \tag{17}\]  
where \(\Delta N_{FSR}\) is the carrier density required to tune the laser by one whole FSR. Approximating using (13) gives:  
\[\frac{c}{2\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}}\approx\frac{mcL_{T}}{2\big{(} n_{eq}L_{eq}+n_{T}L_{T}\big{)}^{2}}\frac{dn}{dN}\Delta N_{FSR} \tag{18}\]  
and using:  
\[\Delta N_{FSR}\approx\frac{I_{FSR}\tau_{N}}{eV} \tag{19}\]  
We obtain a relation for \(I_{FSR}\):  
\[I_{FSR}=\frac{eV\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}}{\Big{(}\frac{dn}{dN} \Big{)}mL_{T}\tau_{N}} \tag{20}\]  
The FM-noise SD at this value of injected current is calculated via (16). Using this value for \(I_{FSR}\), inserting for the constituent terms and simplifying gives:  
\[\Big{|}\Delta\vec{f}\Big{|}^{2}=\frac{\tau_{N}\Big{(}\frac{dn}{dN}\Big{)}mc^{2 }L_{T}^{2}}{VL_{T}^{2}\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}^{3}}=\frac{\tau_{ N}\Big{(}\frac{dn}{dN}\Big{)}mc^{2}}{A_{cs}\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}^{3}} \tag{21}\]  
where \(A_{cs}\) is the cross-sectional area of the laser waveguide section. Interestingly, this result becomes independent of the length of the tuning section if one assumes that \(n_{eq}L_{eq}>>n_{T}L_{T}\). However, the length of the tuning section may play a role in determining \(\tau_{N}\), because \(\tau_{N}\) is carrier density-dependent. Equation (21) can be simplified further by noting that the lasing wavelength \(v_{0}=m\Delta f_{FSR}=mc/\big{(}2\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}\big{)}\). Then, (21) can be written as:  
Figure 6: Temporal trace of the instantaneous laser frequency fluctuations, with the tuning current set to 10 mA and using the parameters in Table 1. The instantaneous frequency can vary by in excess of 50 MHz.  
\[\left|\Delta\hat{f}\right|^{2}=\frac{2\tau_{N}\Big{(}\frac{dn}{dN}\Big{)}v_{0}c}{ A_{cs}\big{(}n_{eq}L_{eq}+n_{T}L_{T}\big{)}^{2}} \tag{22}\]  
This result implies that in order to decrease the FM-noise SD of a tunable laser, one must decrease the carrier lifetime, increase the device area, reduce \(dn/dN\) and increase the length of the laser. The carrier lifetime is bias-dependent, with the lifetime shortening as the bias is increased (see Figure 3b). Hence, the cut-off frequency increases and thus the laser experiences more rapid frequency fluctuations, which will have implications at the receiver of a communications system, with the receiver needing to account for faster (albeit smaller) fluctuations of the lasing frequency. Increasing the lateral area of the guided mode requires that care is taken to avoid two or more resonant axial modes to be guided along the structure. Decreasing \(dn/dN\) may seem counter-intuitive due to the increased shot noise arising from the extra current needed to obtain an increased value of \(\Delta N_{FSR}\). However, since the FM-noise SD depends on \(I_{0}\) and the dependency on the refractive index is \((dn/dN)^{2}\), it is advantageous to allow an increase in the bias current while enjoying a decrease in the overall FM-noise, though larger input currents will heat up the device more and consideration needs to be given to the contribution of thermal tuning, which counteracts against carrier tuning [25; 27]. The only physical parameter that can decrease the \(dn/dN\) in (15) is the effective mass of electrons \(m_{e}^{*}\). This is far from trivial to accomplish, but may be realised through judicious bandgap engineering.