# High Optical Feedback Tolerance of a Detuned DBR Laser for 10-Gbps Isolator-Free Operation
## 4 Discussion  
We finally discuss why wavelength detuning affects feedback tolerance. According to the latest theoretical analysis [13; 14], we infer that the detuned loading effect should be the main reason for the high feedback tolerance of a red-detuned DBR laser, but possibly not all. It is well known that the critical feedback value has a simple and useful analytical expression with some approximations and assumptions from Lang-Kobayashi (L-K) model [11]:  
\[R_{crit}=\frac{R_{a}\mathbf{\tau}_{in}^{2}\mathbf{\gamma}_{r}^{2}}{\mathbf{4}(\mathbf{1}+\mathbf{ \alpha}^{2})(\mathbf{1}-\mathbf{R_{a}})} \tag{1}\]  
where \(\mathbf{R_{a}}\) is the reflectivity of the active region facet, \(\mathbf{\tau_{in}}\) is the roundtrip cavity time, \(\mathbf{\gamma_{r}}\) is the damping rate, and \(\mathbf{\alpha}\) is the effective linewidth enhancement factor. This formula is in good agreement with the experiments in the long external cavity and weak feedback situation. For DBR lasers, except for \(\mathbf{R_{a}}\), three of the four parameters are closely related to wavelength detuning via the detuned loading effect [13; 14]. For a pair of neighboring red-detuned region and blue-detuned region, we ignore the small current difference and assume the same settings except for the detuning. The first parameter \(\mathbf{\tau_{in}}\) is enhanced in both red and blue-detuned regions because the effective roundtrip cavity length is enhanced [14], which is beneficial to feedback tolerance. The next two parameters \(\mathbf{\gamma_{r}}\) and \(\mathbf{\alpha}\) show different trends. In the red-detuned region, \(\mathbf{\gamma_{r}}\) is enhanced and \(\mathbf{\alpha}\) is reduced, so the whole critical feedback value \(\mathbf{R_{crit}}\) is improved. In the blue-detuned region, all the trends are opposite, \(\mathbf{R_{crit}}\) is declined. The detuned loading effect on \(\mathbf{\gamma_{r}}\) and \(\mathbf{\alpha}\) is the main reason for the high feedback tolerance of a red-detuned DBR laser.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Feedback Level (dB)**} \\ \cline{3-8}  & & **W/O OF** & \(-\mathbf{30}\) & \(-\mathbf{20}\) & \(-\mathbf{15}\) & \(-\mathbf{12}\) & \(-\mathbf{10}\) & \(-\mathbf{9}\) \\ \hline \multirow{2}{*}{**Penalty (dB)**} & **BtB** & - & 0.21 & 0.33 & 0.61 & 0.96 & 1.20 & 1.59 \\  & **20 km** & 0.01 & 0.18 & 0.32 & 0.75 & 1.27 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Power penalty at FEC limit of \(1\times 10^{-3}\) for the case of BtB and 20 km fiber transmission.  
Figure 8: Eye diagrams corresponding to the BER results; upper row (**a–d**) is obtained in the BtB configuration under W/O, \(-15\), \(-12\), and \(-9\) dB feedback level; lower row (**e–h**) is obtained in the 20 km transmission configuration under W/O, \(-15\), \(-12\), and \(-9\) dB optical feedback.  
However, when we conduct a rough quantitative estimation, there will be a discrepancy between the experimental and theoretical results. The damping rate of Laser 2 was measured by a Vector Network Analyzer. The maximum occurs at 65 mA with a value of 45 GHz. When we take the largest \(\tau_{\mathbf{r}}\) of 45 GHz, ideally the smallest \(\mathbf{\alpha}\) of 0, longest cavity length of 600-\(\upmu\)m for \(\tau_{\mathbf{in}}\) and \(\mathbf{R_{a}}\) around 30% into the formula, the estimated critical feedback value is \(-15\) dB, which is far lower than over \(-9\) dB obtained in the static test. Therefore, we can recognize that the detuning loading effect is not the only reason for high feedback tolerance, which is identical to the analysis in [13].