# On the Capacity of Optical Backbone Networks
## 3 Channel Capacity
### Capacity of an Optical Channel
Another relevant parameter to characterize the optical channel is spectral efficiency, measured in bit/s/Hz, which is defined as [9; 10]  
\[SE=\frac{C_{ch}}{\Delta v_{ch}}=2\frac{R_{s}}{\Delta v_{ch}}\text{log}_{2}\,(1 +SNR), \tag{10}\]  
where factor 2 stems from the fact that the optical fiber channel supports two optical channels with orthogonal polarizations, commonly referred to as polarization multiplexed (PM) optical channels. By assuming the Nyquist limit, the spectral efficiency \(SE\) can be estimated through closed-form calculations using Equations (5)-(10) by making \(\Delta v_{ch}=R_{s}\). Note that this approximation is widely used throughout this paper because, when using \(\Delta v_{ch}>R_{s}\), the spectral density term \(N_{nli}\) cannot be treated analytically [11], and the closed-form Formula (7) cannot be applied. For this reason, we set \(\Delta v_{ch}=64\) GHz in Table 1, instead of a multiple of 12.5 GHz, as it would be in more realistic scenarios. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Fiber Attenuation Coefficient & \(\alpha\) & 0.22 dB/km \\ Fiber Dispersion Coefficient & \(\beta_{2}\) & \(-21.7\) ps\({}^{2}\)km\({}^{-1}\) \\ Fiber Nonlinear Coefficient & \(\gamma\) & 1.27 W\({}^{-1}\)km\({}^{-1}\) \\ Carrier Frequency & \(\
u_{c}\) & 193.41 THz \\ Carrier Wavelength & \(\lambda_{c}\) & 1550 nm \\ Span length & \(L_{s}\) & 80 km, 100 km \\ EDFA noise figure & \(F_{n}\) & 5 dB \\ Symbol rate & \(R_{s}\) & 64 Gbaud \\ Channel Spacing & \(\Delta v_{ch}\) & 64 GHz \\ Number of Channels & \(N_{ch}\) & 75 \\ WDM bandwidth & \(B_{WDM}\) & 4800 GHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical fiber and system parameters. The obtained results, considering the parameters given in Table 1, are depicted in Figure 3. This figure plots the spectral efficiency as the function of the channel power (\(P_{ch}\)) for different link lengths, considering \(L_{s}=80\) km (Figure 3a) and \(L_{s}=100\) km (Figure 3b). As can be seen, there is a value of the channel power that maximizes the spectral efficiency (\(SE^{max}\)). It can be shown that the value of the optimum launch power per channel is given as [11]:  
\[P_{ch}^{opt}=\sqrt[3]{\frac{N_{ase}}{2\mu_{n}}}. \tag{11}\]  
For \(L_{s}=80\) km we have \(P_{ch}^{opt}=0.95\) dBm and \(SE^{max}=18.3\) bit/s/Hz, while for \(L_{s}=100\) km we have \(P_{ch}^{opt}=2.4\) dBm and \(SE^{max}=16.3\) bit/s/Hz. These results show that \(SE^{max}\) decreases by about 2 bit/s/Hz when the span length increases from 80 km to 100 km because of the increase in the ASE noise power. Another conclusion, we can draw from the figure, is that \(SE^{max}\) also decreases by about 2 bit/s/Hz for every doubling of the link length, and the value of \(P_{ch}^{opt}\) is approximately independent of link lengths. These trends had already been identified in [10]. Figure 3: Spectral efficiency versus channel power for various link lengths: (**a**) span length of 80 km; (**b**) span length of 100 km. Figure 4 shows the variation of the maximum values of spectral efficiency (\(SE^{max}\)) as a function of the total link length. As seen, \(SE^{max}\) decreases in a linear fashion as a function of the link length, when plotted in a logarithmic scale. The spectral efficiency values were computed using Equation (10), which is derived under the hypothesis that the amplitude and phase of the signal at the channel input follow an ideal Gaussian distribution, meaning it is described by a continuous Gaussian constellation (GC). However, in real systems, the input constellations are based on a set of discrete symbols. For a constellation with \(M\) symbols, corresponding, for example, to a modulation format such as PM-MQAM, the ideal spectral efficiency is given by \(SE=2\log_{2}M\) (in bit/s/Hz), where factor 2 accounts for the presence of two polarizations in the channel, as referred to before. Figure 4 also shows the ideal value of \(SE\) for different values of \(M\).