# Comparison of Basic Notch Filters for Semiconductor Optical Amplifier Pattern Effect Mitigation
## 4 ODI vs. MRR Comparison Rationale  
In order to realize the comparison between the ODI and MRR, we must select and assign the proper values to the parameters that critically affect their operation, i.e., (\(\Delta\tau\), \(\Phi_{b,ODI}\)) and (\(R\), \(\Phi_{b,MRR}\)), respectively, so that this comparison is done on a fair basis. For this purpose a rational approach is adopted, which is followed after suitably combining the evidence available and knowledge that have been acquired, both theoretically and experimentally, on these types of modules when employed as notch filters. To this aim, the starting point is to equate the maximum slope of their spectral responses. Taking this as the comparison criterion is dictated by the SOA pattern effect suppression filtering mechanism, according to which the irregularly-shifted spectral components of the amplified pulses are converted by each filter's slope into amplitude changes, which counteract those incurred by the SOA ([27], cf. Figure 4.18). On the other hand, the maximum slope is related to the sensitivity of these filters to input signal perturbations, like those after the SOA, and accordingly to their capability to handle and compensate them [41]. Then, the maximum slope can be determined by finding the zeros of the second derivative of each filter's transfer function and replacing them into the first derivative. This procedure, which is conveniently executed in the angular frequency domain, leads to the equality:  
\[\frac{3\sqrt{3}\cdot r}{8(1-r^{2})}T_{R}=\frac{\Delta\tau}{2} \tag{1}\]  
where the left-hand side gives the maximum slope at critical coupling of the MRR ([42], cf. Equations (3.11)-(3.12) and (B.9)), whose round-trip delay is \(T_{R}=2\pi n_{eff}R/c\), while the right-hand side represents the same quantity for the ODI [15]. Letting then \(r=l=0.95\), so as to ensure that the condition of critical coupling for the MRR is favorably met, but also to preserve the continuity with previous relevant work [33], leads to:  
\[\Delta\tau\cong 12.6T_{R} \tag{2}\]  
which readily gives \(T_{R}=0.8\) ps, or equivalently, \(R\cong 28\) um, for \(\Delta\tau=10\) ps, where the latter value has not been arbitrarily chosen, but according to the selection rules derived in [31] for the conditions under which the ODI should operate to combat the SOA pattern effect. Similarly, the calculated MRR radius complies with the permissible range of values specified for this parameter from the study in [33]. Furthermore, \(\Delta\tau\) and \(R\) are such that the filtered-out data pulses retain their initial shape, as addressed in [15; 29] and verified again below. Having determined the FSR of the ODI and MRR, the next step is to do the same for their detuning. This can be done by applying the criterion of equalization of the red-shifted components of the amplified signal, which must hold in order for the low-pass frequency characteristic of the SOA to be compensated by the high-pass frequency characteristic of the two employed notch filters ([27], cf. Figure 4.4). In this context, the filters' intensity response, \(|H(\lambda)|^{2}_{ODI,MRR}\), is Taylor-expanded to the first order around \(\lambda_{amp}\), thus leading to the following requirement in the linear scale for the filters' slope magnitude, \(K\), [43], which governs the efficiency of converting the uneven SPM-induced spectral broadening to uniform amplified pulse amplitude [44]:  
\[K_{ODI,MRR}=\frac{d|H(\lambda)|^{2}_{ODI,MRR}}{d\lambda}\Bigg{|}_{\lambda= \lambda_{amp}}=\frac{2T_{car}}{\alpha_{LEF}}\frac{\pi c}{\lambda_{amp}^{2}}. \tag{3}\]  
This relationship essentially expresses in mathematical terms the fact that the SOA temporal response, which is limited by the SOA finite carrier lifetime, \(T_{car}\), and the associated pattern effect,can be enhanced by means of the employed filters by a factor that scales at most with the SOA linewidth enhancement factor, \(\alpha_{LEF}\)[45].