# On the Capacity of Optical Backbone Networks
## 4 Link and Network Capacity  
### Link Capacity  
In an abstract way, an optical network can be described as an undirected graph \(G(V,E)\), with \(V=\{v_{1},\ \dots,\ v_{N}\}\) denoting a set of nodes and \(E=\{e_{1},\ \dots,\ e_{K}\}\) denoting a set of links, where \(N=|V|\) is the number of nodes and \(K=|E|\) is the number of links. As mentioned before, in transparent optical networks, all node functionalities take place in the optical domain, and the nodes are built upon ROADMs. Meanwhile, an optical link represents a physical interconnection between two nodes, implemented using optical fibers and optical amplifiers. In bidirectional links, some fibers are used in one direction and others (typically the same number) in the opposite direction. Each optical fiber supports DWDM signals, meaning it carries a specific number of optical channels. For simplicity purposes, we assume that all channels in the links transmit data at the same bit rate. The capacity of an optical link is determined by both the capacity of individual optical channels and the count of optical channels \(N_{ch}\) that it can accommodate. This count is constrained by either the available bandwidth \(B_{WDM}\) or the maximum output power \(P_{max}\) of optical amplifiers, with \(P_{max}\) being limited by the total power \(P_{tot}\) carried by the \(N_{ch}\) channels; that is, \(P_{tot}=N_{ch}P_{ch}\). When considering the Nyquist limit (\(\Delta v_{ch}=R_{s}\)), and if all the channels have the same modulation format and symbol rate, one can arrive by applying (10) and (12) to the following expression for the link capacity:  
\[C_{link}=2N_{ch}R_{s}\text{log}_{2}(1+SNR), \tag{14}\]  
where \(SNR\) is evaluated through Equations (6)-(8). Figure 5 plots the link capacity as a function of the channel power for four different symbol rate values. The primary conclusion drawn from this figure is that the maximum link capacity (\(C_{link}^{max}\)) remains constant regardless of the symbol rate. However, this maximum capacity value shifts to higher power levels with increasing symbol rates. This trend is independent of the link length, although the maximum link capacity decreases when the link length increases. As seen, for \(L=400\) km, \(C_{link}^{max}=65.6\) Tbit/s, while for \(L=800\) km, we have \(C_{link}^{max}=56.1\) Tbit/s. Table 5 presents several parameters to provide insight, including the optical channel power value that maximizes the link capacity (\(P_{ch}^{opt}\)) for each symbol rate. Interestingly, the total optical power value remains the same for all the symbol rates. This value is equal to 19.6 dBm, which is well within the capabilities of commercial EDFAs. Furthermore, Table 6 presents some experimental results, reported in the literature, for both the channel and links capacities at various symbol rates. In particular, the results for a symbol rate of 128 Gbaud are quite impressive, revealing values of 1.49 Tbit/s and 52.1 Tbit/s for the channel and links capacities, respectively. However, a drawback of these results is the achieved distance of only 80 km. \begin{table}
\begin{tabular}{c c c c} \hline
**Modulation** & \begin{tabular}{c} **Data Bit Rate** \\ **(Gbit/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Reach (km)** \\ **\(L_{s}\) = 80 km** \\ \end{tabular} &
\begin{tabular}{c} **Reach (km)** \\ **\(L_{s}\) = 100 km** \\ \end{tabular} \\ \hline PM-BPSK & 100 & 9500 & 6000 \\ PM-QPSK & 200 & 4750 & 3000 \\ PM-8 QAM & 300 & 1650 & 1050 \\ PM-16 QAM & 400 & 1000 & 650 \\ PM-32 QAM & 500 & 400 & 250 \\ PM-64 QAM & 600 & 250 & 150 \\ PM-128 QAM & 700 & 100 &......... \\ \hline \end{tabular}
\end{table} 
Table 4: Optical reach values (\(@BER_{pre-FEC}=10^{-3}\)).