# Spectrum Allocation Using Integer Linear Programming and Kerr Optical Frequency Combs
## 2 Materials and Methods  
This work proposes a spectrum allocation method based on constrained optimization using Integer Linear Programming (ILP), referred to as RMISA-ILP-OFC, to maximize the amount of allocated spectrum. This approach utilizes the optical network proposed in [19], which employs optical microresonators of 113 \(\upmu\)m and 450 \(\upmu\)m to achieve optical frequency combs (OFCs) with free spectral ranges (FSRs) of 12.5 GHz, 50 GHz, 100 GHz, and 200 GHz, in compliance with the International Telecommunication Union (ITU) recommendations. Although this methodology is proposed, it can be easily reconfigured to use other types of optical sources, such as electro-optic modulators, semiconductor lasers, or gain-modulated microcavities. This flexibility is due to the approach's consideration of key physical characteristics like transmission distance, Optical Signal-to-Noise Ratio (OSNR), and frequency spacing. These features make it compatible with various Wavelength Division Multiplexing (WDM) sources, allowing the network to adapt seamlessly to the needs of operators and existing infrastructure, ensuring a scalable and future-proof design. We use a programmable pulse shaper to physically assign one or more wavelengths, following the instructions of the ILP algorithm. This algorithm adjusts the modulation level in QAM multilevel modulators to optimize performance based on link characteristics, such as transmission distance. The ILP approach ensures efficient wavelength allocation tailored to the specific requirements of the network request. This algorithm plays a central role in dynamic spectrum management, allowing the adaptation of carriers based on transmission distance and bandwidth demands, ensuring efficient use of the available spectrum, and enabling the creation of a flexible network that adapts to changes in bandwidth and transmission distance. Figure 1 shows the topology of the proposed network. It is important to highlight that the pulse shaper can assign an individual wavelength or a range of them in the case of a super-channel, enabling the creation of a flexible grid, a fundamental aspect of our methodology. To generate the OFCs with the proposed spacings, we use two structures based on MRRs and the configuration from [19]. The first structure uses four MRRs with an FSR of 50 GHz, where thermal control adjusts the position of the OFCs to obtain a new FSR of  
Figure 1: Scheme for spectrum allocation adaptation from [18]. 12.5 GHz. The optical combs generated by the rings are combined using an interleaver, forming a final optical comb with the desired specifications. On the other hand, the second structure uses two MRRs with a free spectral range (FSR) of 200 GHz and follows the same approach to achieve a final line spacing of 100 GHz. Figure 2 shows the schematic of these MRRs. To obtain the optical signal from the microresonators, we utilize the Lugiato-Lefever Equation (LLE) as proposed in [19], because it allows us to simulate the optical generation of frequency combs based on intrinsic characteristics, such as the Kerr coefficient and the loss coefficient of the MRR. The Lugiato-Lefever equation (LLE) is defined as follows:  
\[t_{R}\frac{\partial E(t,\tau)}{\partial t}=\left[-\alpha-i\delta_{0}+iL\sum_{k \geq 2}\frac{\beta_{k}}{k!}\left(i\frac{\partial}{\partial\tau}\right)^{k}+i \gamma L|E|^{2}\right]E+\sqrt{\theta}E_{in} \tag{1}\]  
where \(\delta_{0}\) represents the laser detuning and \(E_{\text{in}}\) is the pump field. To solve this equation numerically, we employ the split-step Fourier method. It is important to emphasize that the pump power and detuning are normalized (see Equations (2) and (3) below). \[\Delta=\delta_{0}/\alpha \tag{2}\]  
\[S=E_{in}\sqrt{\gamma L\theta/\alpha^{3}} \tag{3}\]  
To carry out the spectrum allocation process, we propose an ILP algorithm. This approach prioritizes customer allocation using the cost equation outlined in (4).