# On the Capacity of Optical Backbone Networks
## 3 Channel Capacity
### Capacity of an Optical Channel
The crossing points between the modulation's spectral efficiency and the Gaussian constellation's spectral efficiency enable the evaluation of an upper bound on the maximum reach achieved for each set of symbols (see Table 2). Figure 4: Maximum value of the spectral efficient (\(SE^{max}\)) as a function of the total link length, considering a Gaussian constellation (GC) and other discrete constellations with different numbers of symbols (\(M\)): (**a**) span length of 80 km; (**b**) span length of 100 km. The results of Table 2 clearly evidence the trade-off between the cardinality of the constellation (number of symbols) and the maximum reach; as the number of symbols increases, reach decreases significantly. For example, one observes a reach reduction between 75% to 80% when the number of symbols quadruple. This reduction increases further to about 95% when the number of symbols increases 16 times. The values of the maximum reach also decrease when the span length increases. By moving from \(L_{s}\) = 80 km to \(L_{s}\) = 100 km, one observes a reach reduction of about 37%. It is also worth mentioning the fact that the results given in Table 2 are closer to the results of Figure 2 of [25], despite these results having been obtained with a more rigorous approach. The current SE record of 17.3 bit/s/Hz was obtained using a modulation format with 4096 symbols and polarization multiplexing (PM-4096 QAM) over 50 km [26], which is quite close to the value of 18 bit/s/Hz shown in Figure 4a) for a length of 80 km. Another remarkable experimental result was the achievement of a SE of 14.1 bit/s/Hz at a reach of 500 km using PM-256-QAM) [27]. These two experimental results confirm the previously mentioned trend: a reduction in the reach by approximately 90% when the number of symbols increases by a factor of 16. According to Equation (10), the optical channel capacity is related to the channel spacing \(\Delta v_{ch}\), which permits us to write  
\[C_{ch}=SE\times R_{s}(1+\beta). \tag{12}\]  
Table 2 also presents values for the channel capacity obtained using this equation, considering the Nyquist limit \(\beta=0\). From Equation (12) it can be concluded that two strategies can be employed to increase \(C_{ch}\): (1) increasing the spectral efficiency; (2) increasing the symbol rate. The first strategy suffers from the limitations of spectral efficiency already referred. In this way, it is expected a huge reach reduction for increasing values of the capacity. On the other hand, the second strategy increases the sensitivity to noise and nonlinearities and consequently also reduces the reach. However, this reduction can be compensated for by increasing the channel power so that in the end, we only experience a modest decrease in the reach for higher capacity values. The reason for this behavior is that by increasing the channel power in the same proportion as the symbol rate, the power spectral density \((P_{ch}/R_{s})\) is kept constant, and in this way, the NLI power does not undergo any change (see Equations (7) and (8)). To give more insights into the problem, let us analyze what happens if we double the channel capacity, starting, for example, with a capacity of 200 Gbit/s based on a PM-QPSK scheme with a symbol rate of 64 Gbaud. By using the first strategy, it is necessary to double the spectral efficiency by going from PM-QPSK (4 bit/s/Hz) to PM-16 QAM (8 bit/s/Hz) in order to achieve 400 Gbit/s. However, the last modulation scheme is more sensitive to both noise and nonlinearities, requiring as a consequence a SNR 6.8 dB higher (see Table 3). Therefore, the number of spans supported by PM-16 QAM is approximately 4.8 times smaller than that supported by PM-QPSK, which translates into a reach reduction of about 80%, in line with the results given above. Alternatively, we can go to 400 Gbit/s by doubling the symbol rate to 128 Gbaud and keeping the modulation PM-QPSK. In this case, the 50% reduction in reach due to the increases in the noise power (see Equation (6))