# Polynomial Modeling of Noise Figure in Erbium-Doped Fiber Amplifiers
## 2 State-of-the-Art
### Comparison of Different EDFA Models  
Estimating the parameters of network elements like EDFAs using specialized models is crucial for the optimal design, operation, and maintenance of optical communication systems. This estimation allows for accurate performance predictions of network elements under various conditions without the need to deploy the network initially, thereby greatly aiding in network design. Various models for EDFAs have been proposed in numerous articles [7; 10]. Most early models focus on gain estimation, which is notoriously difficult to analyze due to two primary factors: static-state inter-channel gain variation and dynamic gain excursion [17]. Static-state variation arises from uneven gain across channels due  
Figure 2: Illustration of \(P_{ASE_{AMP}}\). The output spectrum (red) is shown alongside the input spectrum (blue) adjusted by the amplifier gain (blue dashed). \(P_{ASE_{AMP}}\) represents the noise introduced by the EDFA to the amplified SSE level. to the dependence of the fiber's absorption and gain spectra on the signal wavelength. Dynamic gain excursion refers to rapid gain fluctuations in response to changes in input power or channel configurations, which are common in optical networks [7]. Analytical models derive formulas for the theoretical estimation of EDFA parameters, such as gain and noise figure. By starting from fundamental physical equations, such as those governing the carrier populations in the energy levels of erbium ions, comprehensive analytical expressions for these parameters can be obtained. The main advantage of analytical models is that they depend solely on a priori knowledge of the equipment and a restricted amount of measurements, with values possibly provided by manufacturers, and do not require any deployment or input-output characterization. While this is the simplest method, it does come with several challenges. Firstly, these models are typically very complex and rely on several assumptions, such as the two-energy-level approximation for erbium ions [20], which can compromise their accuracy. Additionally, they do not consider dynamic effects, such as those previously mentioned for gain estimation, nor manufacturing defects. Another issue is the inconsistency between different devices. EDFAs can be constructed with various architectures and include different components in their schematics. Consequently, an analytical model based on the underlying physical equations must be adapted for each device architecture, making it impractical to apply the same model to different brands and devices. This issue is directly related to another challenge with commercial devices: manufacturers often keep the details of their device architectures confidential, making it difficult to develop accurate models for consumers. Some simpler analytical models for gain estimation are available, such as those based on a center of mass function [21], which enable evaluation of gain in AGC mode using a straightforward expression. These models can be easily adapted to multiple devices. However, their accuracy is limited because they overlook various internal effects of the devices. Specifically, they assume an ideally constant gain spectrum shape, which does not hold true for real devices, and they focus on the impact of the loaded channel configuration, disregarding gain variations due to channel power fluctuations. Finally, there is the issue of aging. The physical parameters used in these models change over time, necessitating continuous updates to maintain their reliability. This process can be very costly, as these parameters are determined through complex experimental measurements. While it is possible to incorporate time-dependent factors into the equations, doing so would further increase the already high complexity of the model. Given the challenges associated with the analytical models of EDFA parameters, research has explored alternative solutions and technologies. The most promising alternatives are machine-learning models, especially those based on neural networks.