## 4 Discussion  
The existing mature photonic integration fabrication processes are very flexible with respect to device geometry. They are however limited by a lack of commercially available on-chip isolators, and thus new approaches are required to minimize the effect feedback has on laser stability. The current work proposes a theoretical extension of laser dynamics under EOF by considering two external reflections, one from each side of the cavity. The proposed analysis yields new laser dynamic equations. These are numerically solved, which show the existence of a stable regime with high feedback tolerances. Similar results have been seen in previous double external cavity schemes [28], with EOF on the same side of the laser cavity, which showed robust stabilization of the chaotic regime by tuning the external feedback parameters. Yet other feedback regimes are not explored, and neither analytical stability conditions nor dependence with free laser running frequency are reported. The present analysis shows that feedback-insensitivity is achievable for a wide range of feedback strengths, under conditions that can be met with current laser fabrication processes and components, such as phase shifters for meeting condition (12) and couplers for condition (8). It has previously been shown, for the single feedback case, that tuning the feedback phase can result in linewidth narrowing [13], however this still requires low feedback levels and a precise phase shift which can suffer variations due to external parameters, such as temperature or driving currents. In the approach proposed in this work, the stability conditions do not require specific values, instead relating the feedback parameters from one side to those from the other side. For example, condition (11) only implies equal round-trip times at the external cavities, regardless of value. This allows for additional flexibility in the feedback parameters and gives versatility to the device. Furthermore, this method allows for feedback-insensitivity across the full spectra, which is not seen in the single feedback case. This is of particular importance for tunable lasers, as all lasing frequencies are thus equally affected. Additionally, it relaxes the need for an isolator, reducing the cost and size of packaging processes. Another significant improvement of the proposed method with respect to the single feedback case is the increase in feedback tolerance: by choosing feedback parameters close to the stability conditions of Equations (11) and (12), higher levels of feedback strength are allowed without seeing multi-valued solutions, which results in experimentally seen mode-hopping. As a weak feedback approximation is used, the upper bound for feedback tolerance cannot be extracted from this analysis. Despite this, even under this approximation, tolerances are higher than that of the single feedback case. Furthermore, this system has a high tolerance to deviations from the optimal stability conditions as analyzed in the previous section. Linewidth narrowing can be achieved in these cases for certain frequency values, which can be tuned by selecting appropriate feedback parameters, as was the case for single feedback conditions, while maintaining stable solutions. Finally, while the dynamics under consideration are complex, the laser system itself involves a straight-forward configuration using widely used on-chip components, which are available in generic foundry platforms. Previously studied methods to reduce feedback sensitivity include resourceful yet intricate designs. The proposed system is, in contrast, potentially easier to design, fabricate and characterize. An experimental study of this laser system is essential to validate the obtained results, and more importantly to explore the limitations of the model, and is the next step for a more comprehensive understanding of the proposed system.