## 3 Network Model
### Flexible Modulation Format  
In [48; 49], four different types of SChs operate with the same MF throughout their routing paths. The relationship between traffic transmission distance, MFs, and network resource utilization is not discussed in the two papers. On the one hand, since traffic rates from 100 gigabit per second (Gb/s) to 10 Tb/s and even higher can be supported in SCNs, lower-order MFs implemented with 28-Gbaud or 56-Gbaud transceivers cannot meet the requirements of high-bit-rate traffic transmission. On the other hand, although the traffic rate can be increased by using higher-order MFs, the transmission distance will be reduced as a result. Therefore, it is a challenge for SCNs to select the appropriate MFs for traffic requests based on traffic rate and routing path length. According to the analysis in [17], the transmission distance of add/drop optical signals utilizing WXCs in SCNs is no less than that in conventional optical network architectures, and the transmission distance of the spatially bypassed optical signals utilizing SXCs (since SXCs are cost-effective) is extended by 98%. Thus, by analyzing the relationship between the bit rate of OChs, the transmission distance of OChs, and MFs in WDM networks, we can derive their relationship in SCNs and obtain the adoption method of MFs. In this paper, the derivation was finished by studying dispersion-uncompensated optical signal transmission based on optical orthogonal frequency division multiplexing and Nyquist-WDM. In addition, we use dual-polarization to double the spectral efficiency. Assumptions:  
* The transmission link is composed of uniform and homogeneous fiber spans, all of which have the same physical characteristics;
* Seamless multiplexing of signals with the same MF is achieved on the same transmission link;
* Same forward error correction (FEC) scheme with higher coding gain is adopted on the same transmission link;
* Signal distortion caused by fiber nonlinearity is Gaussian distributed;
* Signal-to-noise nonlinear interaction is negligible;
* Ideal digital coherent transceivers are used. In this system, the optical signal-to-noise ratio (OSNR) can be expressed as:  
\[\text{OSNR}=\frac{I\exp\left(-(I/I_{0})^{2}\right)}{2n_{0}+I\left(1-\exp\left( -(I/I_{0})^{2}\right)\right)}, \tag{1}\]  
where \(I\) is the launch power spectral density of the transmitted signal in the presence of nonlinear interference, \(I_{0}\) is the nonlinear characteristic power density, and the noise is the sum of the amplified spontaneous emission noise and the four-wave mixing noise [50], and  
\[I_{0}\equiv\frac{1}{\gamma}\sqrt{\frac{8\pi\alpha|\hat{\rho}_{2}|}{3N_{s}h_{e} ln(B/B_{0})}} \tag{2}\]  
The Q-factor used to measure the system performance is introduced, which is considered equal to the OSNR when QPSK is used. The maximum available Q-factor (Q\({}_{\text{m}}\)) and the maximum effective OSNR (OSNR\({}_{\text{m}}\)) are obtained by differentiating Q to \(I\):\[\text{Q}_{\text{m}} =\text{OSNR}_{\text{m}}=\frac{1}{3}\bigg{(}\frac{I_{0}}{n_{0}} \bigg{)}^{\frac{2}{3}}=\frac{(8\pi\alpha|\beta_{2}|)^{\frac{1}{3}}}{3[3n_{0}^{2} \gamma^{2}N_{s}h_{e}\ln(B/B_{0})]^{\frac{1}{3}}} \tag{3}\]  
Moreover, in the model, (1) the power spectral density of ASE noise, \(n_{0}\), is proportional to \(N_{s}\cdot 10^{\alpha L_{s}}\); (2) the use of ideal digital coherent transceivers leads to \(h_{e}\approx 1\); (3) the modulation format order \(n\) needs to be considered when solving for the \(\text{OSNR}_{m,D}\) and the \(\text{Q}_{m,D}\) due to the use of dual-polarization.