# Analysis of Available Components and Performance Estimation of Optical Multi-Band Systems
## 3 Quantifying the Capacity and Transparent Reach of an OMB System
### Physical Layer formalism
 \tag{5}\]  
\[\begin{array}{l}\text{where }z_{1}~{}=~{}\big{(}\frac{2}{a}\big{)}^{2}+2L^{2} \big{(}{N_{s}}^{2}-1\big{)}/\bigg{(}\sum\limits_{k=x_{1}}^{x_{2}}\frac{1}{1+ \left(2k\uppi/\left(al\right)\right)^{2}}\bigg{)}^{2},~{}z_{2}~{}=~{}\big{(} \frac{2}{a}\big{)}^{2}+2L^{2}\big{(}{N_{s}}^{2}-1\big{)},\\ x_{1}~{}=~{}-\frac{\lambda^{2}B^{2}DLN_{s}{}^{2}}{16c},~{}x_{2}~{}=~{}\frac{ \lambda^{2}B^{2}DLN_{s}{}^{2}}{2c},~{}\gamma\text{ is the non-linear fiber coefficient, which takes the value of 1.31744 W\({}^{-1}\cdot\)km\({}^{-1}\cdot\)km\({}^{-1}\), \(D\) is the local dispersion, and \(a\) is the fiber attenuation parameter, which are different in each band and can be extracted from Figure 2 of [20]. Table 6 illustrates the a, \(D\), and \(\lambda\) for the central wavelength of each band. Next, \(L\) is the span length, \(N_{ch}\) is the number of channels of the band, \(B\) is the channel bandwidth, and \(B+GB\) is the channel spacing, which is equal to 37.5 GHz in this work. Finally, \(\Phi\) is a parameter that depends on the modulation format and takes the value of 1 for QPSK, 17/25 for 16 QAM, and 13/21 for 64 QAM. The next step is to calculate the power of the channels in each band at the beginning of a link, in a way that ensures a balanced OSNIR performance across all channels at the end of the link. For this purpose, we employed closed-form expressions instead of more complex methods, e.g., Split Step Fourier Method, in order to perform power optimizations in a reasonable-time manner. The power arrangement is an important procedure, as it can minimize the impact of SRS, which as an inter-band effect can deplete the power of the channels in the lower bands degrading significantly their OSNIR. At the same time, it can also increase the channel power in the higher bands, stressing them toward the non-linear regime, where the increased FWM can severely degrade their OSNIR. As a result, a collective optimization of the OSNIR performance across all bands is required. For this purpose, we use the method of [23], which minimizes a suitably chosen merit function subject to specific power constraints \(P_{min}\) and \(P_{max}\) as follows:  
\[ \begin{array}{c}A\left(P_{S_{1}},P_{S_{2}},P_{C},P_{L}\right)=\sum\limits_{ b}\left(\frac{1}{\text{OSNIR}_{b}\left(P_{S_{1}},P_{S_{2}},P_{C},P_{L}\right)} \right)^{2}\\ \text{minimize }A\left(P_{S_{1}},P_{S_{2}},P_{C},P_{L}\right)\\ \text{subject to }P_{min}\leq P_{S_{1}},P_{S_{2}},P_{C},P_{L}\leq+1\text{ dBm}\end{array}  \tag{6}\]  
where \(P_{S_{1}},P_{S_{2}},P_{C},P_{L}\) are the wanted quantities, which represent the power of the middle channel of S\({}_{1}\), S\({}_{2}\), C, and L-bands at the beginning of each link. \(P_{min}\) is set in order to avoid a catastrophic OSNIR degradation due the small channel power, while \(P_{max}\) is set to e.g., \(+1\) dB, to avoid a DFA operation in its saturation regime, which can result in insufficient amplification gain and consequently to a severe OSNIR degradation. In essence, Equation (6) minimizes the OSNIR difference between the various exploited bands, providing a more balanced OSNIR performance across the entire spectrum. Using this method, the appropriate values of each band are calculated, allowing each band to swing between the ASE-limited and NL-limited regime, mitigating in this way the catastrophic SRS impact on the S-band. In this work, we calculate the transmission performance for a system where all bands are fully loaded with channels, which is the worst possible scenario in terms of physical layer performance.