# On the Capacity of Optical Backbone Networks
## 3 Channel Capacity
### Capacity of an Optical Channel  
As previously mentioned, an optical channel can be seen as a communication pathway for transmitting information in the optical domain from a sender to a receiver, using an optical fiber as a transmission medium. This channel is characterized by its carrier frequency \(\
u_{c}\) in Hz and its occupied bandwidth \(B_{ch}\) in Hz, whose minimum value is equal to \(R_{s}\), as discussed earlier. The capacity of an optical channel is defined as the maximum data rate at which the information can be effectively transmitted through the channel. This capacity is typically expressed in bit/s. Equation (4) can also be applied to compute the capacity of an optical channel, denoted as \(C_{ch}\), under the assumption that the noise sources present in these channels are modelled as AWGN sources. One important noise source in optical systems is ASE noise. This noise is generated inside of optical amplifiers simultaneously with signal amplification and can be effectively described by a random optical field with statistical properties like those of AWGN noise [9]. The PSD of the ASE noise at the end of a chain of \(N_{s}\) amplifiers, spaced by fiber spans of length \(L_{s}\), is given by  
\[N_{use}=N_{s}h\
u_{c}f_{n}(a_{s}-1), \tag{5}\]  
where \(h\) is [PERSON]'s constant (in joule-second), \(f_{n}\) is the noise figure (\(f_{n}=10^{F_{n}/10}\), with \(F_{n}\) in dB), and \(a_{s}=10^{A_{s}/10}\). NLI is another significant noise source caused by the Kerr effect in optical fibers, as seen before. Interestingly, it has been demonstrated in [23] through simulations and experiments that the impact of NLI noise on WDM links, supported in dispersion uncompensated fibers, can also be modeled as additive Gaussian noise. Furthermore, it was shown in [11] that under specific conditions, such as the Nyquist limit, the white noise assumption leads to quite accurate results. Note that such a limit is achieved when all the WDM signal channels have a rectangular spectral width and a frequency spacing \(\Delta\
u_{ch}\) equal to \(R_{s}\).This permits the characterization of the NLI noise also as an AWGN process with power spectral density of \(N_{nili}\). As the ASE and NLI noises are assumed to be uncorrelated, their power spectraldensities simply add, resulting in \(N_{0}=N_{\text{ase}}+N_{nili}\). In these circumstances, the received signal-to-noise ratio for a given optical channel can be described as  
\[SNR=\frac{P_{ch}}{(N_{\text{ase}}+N_{nili})R_{s}}, \tag{6}\]  
where \(P_{ch}\) denotes the average output optical power per channel, which is assumed to be equal to the input power, since all losses are compensated for by optical amplifiers. A rigorous characterization of \(N_{nli}\) is not an easy task, and many studies have been published on this topic (see, for example, [11; 23]). Fortunately, some closed-form approximations have also been published [10; 11; 24], which facilitates the evaluation of \(N_{nli}\). One of these approximations, which is based on the white noise assumption, allows us to write the PSD of the NLI at the end of a fiber link with \(N_{s}\) spans in the following way:  
\[N_{nli}=\mu_{n}P_{ch}^{3}, \tag{7}\]  
where \(\mu_{n}\) is the NLI coefficient given by  
\[\mu_{n}\approx\left(\frac{2}{3}\right)^{3}N_{s}\gamma^{2}L_{ef}\frac{\text{ln} \Big{(}\pi^{2}|\beta_{2}|L_{ef}B_{WDM}^{2}\Big{)}}{\pi{R_{s}}^{3}\Big{|}\beta_ {2}\Big{|}}, \tag{8}\]  
with the \(L_{ef}\) being the span effective length given as  
\[L_{ef}=(1-\exp(-2a_{N}L_{s}))/(2\alpha_{N}), \tag{9}\]  
where \(L_{s}\) is the span length and \(a_{N}\) is the fiber attenuation coefficient in Np/km, i.e., \(\alpha_{N}=\alpha_{\text{dB/km}}\)\(/20\text{log}_{10}\epsilon\).