# 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  
In an OMB system, the main physical layer impairments are ASE noise, FWM, and SRS [19, 20, 21, 22, 23]. ASE noise and FWM are intra-band effects, while SRS is an inter-band effect, as it leads to a power exchange from the lowest bands, such as the S-band, toward the highest ones, such as the L-band. By making the assumption that FWM is a Gaussian Noise (GN) source, statistically independent from ASE noise, then, we can calculate the OSNIR as follows [20]: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Band** & **Used Range (nm)** & **Range (nm)** & **Number of Channels** & **Noise Figure (dB)** & **Amplifier Type** \\ \hline S\({}_{1}\) band & 1455–1480 & 25 & 92 & 5.5 & TDFA \\ S\({}_{2}\) band & 1485–1510 & 25 & 89 & 5.5 & TDFA \\ C band & 1530–1565 & 35 & 116 & 5.5 & EDFA \\ L band & 1570–1615 & 45 & 141 & 6.0 & EDFA \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Spectral partitioning used in our study. Figure 2: The transmission link used to interconnect two consecutive optical nodes. \[\text{OSNIR}=\frac{P_{ch}\cdot\prod\limits_{i=1}^{N_{s}}G_{SRS,i}}{P_{ASE}+P_{FWM}} \tag{1}\]  
where \(P_{ch}\) is the power of the examined channel, \(P_{FWM}\) is the power of FWM interference, \(G_{SRS,i}\) is the SRS gain/loss effect of the channel under observation at the \(i\)th fiber span, and \(P_{ASE}\) is the power of ASE noise. The OSNIR can be directly used to calculate the BER for various modulation formats as follows [24]:  
\[ \begin{array}{l}\text{BER}_{\text{QPSK}}=\frac{1}{2}erfc\bigg{(}\sqrt{\frac {\text{OSNIR}}{2}}\bigg{)}\\ \text{BER}_{\text{16 QAM}}=\frac{3}{8}erfc\bigg{(}\sqrt{\frac{\text{OSNIR}}{ 10}}\bigg{)}\\ \text{BER}_{\text{64 QAM}}=\frac{7}{24}erfc\bigg{(}\sqrt{\frac{\text{OSNIR}}{ 42}}\bigg{)}\end{array}  \tag{2}\]  
Next, the power of ASE noise is given by  
\[P_{ASE}=\sum\limits_{i=1}^{N_{s}}\Bigg{[}hf(NF_{i}\cdot G_{i}-1)B_{0}\prod \limits_{r=i+1}^{N_{s}}G_{SRS,r}\Bigg{]} \tag{3}\]  
where \(G_{i}\) is the amplifier gain and \(NF_{i}\) is the noise figure at the \(i\)th amplification stage. \(N_{s}\) is the number of fiber spans a channel is traversing before its power is equalized using e.g., a WSS. The SRS gain/loss effect for the \(i\)th wavelength in the \(i\)th fiber span, is given by [20,25]:  
\[G_{SRS,i}=P_{tot,SRS}\frac{e^{\frac{g^{\prime}B\cdot L_{eff}}{2\mathcal{L}_{ e}}(j-1)P_{tot,SRS}}}{\sum\limits_{m}\Bigg{[}P_{m,0}e^{\frac{g^{\prime}B\cdot L_{ eff}}{2\mathcal{L}_{e}}(m-1)P_{tot,SRS}}\Bigg{]}} \tag{4}\]  
where \(g^{\prime}\) is the Raman gain slope equal to \(4.9\times 10^{-27}\) m/(W\(\cdot\)Hz), \(A_{e}\) is the effective cross-sectional area of the fiber equal to 80 \(\upmu\)m\({}^{2}\), and \(P_{m,0}\) is the power of the \(m\)th interfering channel at fiber input. The term \(P_{tot,SRS}\) sums the power of channels that interact within the SRS gain bandwidth, which in our work is considered as 15 THz, since we use the triangular approximation [26]. This wide SRS gain bandwidth makes the OSNIR of one band a function of the power level of the channels in all other bands. To estimate the impact of FWM, we use the expression of \(P_{FWM}\) in [27], which was shown in [19] to be in a very good agreement with numerical results across the S, C, and L-bands:  
\[ \begin{array}{l}P_{\text{FWM}}=\frac{32}{27}\frac{\gamma^{2}L_{eff}{}^{2}P_ {0}{}^{3}N_{s}{}^{2}c}{\lambda^{2}B^{2}D\sqrt{z_{1}}}\bigg{(}1+\frac{4e^{-al} }{\left(1-e^{-al}\right)^{2}}\bigg{)}\big{(}asinh\bigg{(}\frac{\uppi\lambda^{2 }DB^{2}}{8c}N_{ch}\frac{2b}{\uppi^{2}CB}\sqrt{z_{2}}\bigg{)}-\frac{5}{3}\Phi Log \Big{(}N_{ch}\frac{B}{\uppi^{2}CB}\Big{)}\big{)}\\ -\frac{32}{27}\frac{\gamma^{2}L_{eff}{}^{2}P_{ch}{}^{3}N_{s}{}^{2}c}{\lambda^{2 }B^{2}D\sqrt{z_{1}+12L^{2}}}\Big{(}\frac{4e^{-al}}{\left(1-e^{-al}\right)^{2} }\Big{(}asinh\bigg{(}\frac{\uppi\lambda^{2}DB^{2}}{8c}N_{ch}\frac{2b}{\uppi^{ 2}CB}\sqrt{z_{2}+12L^{2}}\bigg{)}-\frac{5}{3}\Phi Log\Big{(}N_{ch}\frac{B}{ \uppi^{2}CB}\Big{)}\Big{)}\end{array}