# Polynomial Modeling of Noise Figure in Erbium-Doped Fiber Amplifiers
## 2 State-of-the-Art
### Noise Figure of EDFAs  
EDFAs introduce unwanted optical power fluctuations to the amplified output signal, with ASE noise being the most significant contributor, as discussed earlier. This unwanted addition reduces the OSNR of the transmitted signal, making its measurement essential. To quantify this effect, a parameter known as the noise figure is used, similar to the approach in electronic amplifiers [17]. The noise figure NF is defined as the noise factor, F, expressed in dB units. \[NF=10\,log_{10}\,F \tag{1}\]  
The noise factor, F, is the ratio of the Signal-to-Noise Ratio (SNR) at the input and output of the amplifier. It depends on the optical frequency, \(\
u\), and the baseband frequency, \(f\), of the electrical output from the photodetector used in the receiver. \[F\left(\
u,f\right)=\frac{SNR_{IN}}{SNR_{OUT}\left(\
u,f\right)} \tag{2}\]  
In essence, the noise figure indicates the degradation of signal quality due to the amplifier's insertion into the optical line. Historically, from the 1980s to the early 2000s, it was believed that the noise figure of a high-gain, phase-insensitive linear amplifier had a lower bound of 3 dB [18], due to the Heisenberg uncertainty principle. Even though the noise figure of a phase-sensitive amplifier can be below 3 dB, it requires the transmission of a phase-conjugated signal and idler with a pump reference, and this remains a research topic without commercial deployment [19]. When discussing lumped amplifiers like EDFAs, however, the 3 dB noise figure limit is fundamental, and achieving this value is still considered a significant challenge. The best result obtained in the measurements used in this work is around 3.7 dB. To evaluate the noise figure, we consider the setup shown in Figure 1. The receiver is assumed to be ideal, and the laser source is shot-noise limited, meaning it has a high SNR and that the primary contribution to source noise is shot noise. The SNR for a shot-noise-limited source is given by:  
\[SNR_{IN}=\frac{\eta P_{IN}}{2\hbar\
u B_{e}} \tag{3}\]  
where \(\eta\) is the quantum efficiency of the photodetector, \(h\) is [PERSON]'s constant expressed in J - s, and \(B_{e}\) is the noise bandwidth in Hz. For an ideal receiver, \(\eta\) equals 1. The output SNR of the amplifier can be expressed as:  
\[SNR_{OUT}=\frac{G^{2}P_{IN}^{2}}{B_{e}[S_{e}\left(\
u,f\right)+\eta^{-1}S_{ shot}]} \tag{4}\]  
where the noise spectral density, \(S_{e}\), expressed in \(\mathrm{W^{2}\,Hz^{-1}}\), accounts for all noise contributions except shot noise. The noise factor is then:  
\[F\left(\
u,f\right)=\frac{S_{total}\left(\
u,f\right)}{2\hbar fG^{2}P_{IN}} \tag{5}\]  
The spectral density of signal-spontaneous beating noise and shot noise can be written as:  
\[S_{sig-sp}=4\rho_{ASE}GP_{IN},\ S_{shot}=2\hbar fGP_{IN} \tag{6}\]  
where \(\rho_{ASE}\) is the ASE power density. Other noise contributions can be neglected if the input channel power is above the effective input noise of the amplifier (typically around \(-50\,\mathrm{dBm}\)) [17]. The noise factor can now be rewritten as:  
Figure 1: Input and output SNR measurement setups for noise figure evaluation. \[F=\frac{4\rho_{ASE}GP_{IN}+2 hfGP_{IN}}{2 hfG^{2}P_{IN}}=\frac{2\rho_{ASE}}{hfG}+\frac{ 1}{G} \tag{7}\]  
where the term \(1/G\), corresponding to the shot noise, can usually be neglected. Finally, the expression can be rewritten by replacing \(\rho_{ASE}\) with the corresponding power, a metric more commonly obtained from spectrum analyzer measurements. This is achieved by multiplying the ASE noise power density by the corresponding noise bandwidth, \(B_{0}\). The final expression for the noise factor is:  
\[F=\frac{2\rho_{ASE}B_{0}}{hfGB_{0}}=\frac{P_{ASE_{AMP}}}{hfGB_{0}} \tag{8}\]  
The term \(P_{ASE_{AMP}}\) represents the ASE power introduced by the amplifier. It is calculated by measuring the ASE power at the output of the amplifier and subtracting the Source Spontaneous Emission (SSE) power multiplied by the gain of the amplifier.