# Lossless WDM PON Photonic Integrated Receivers Including SOAs
## 3 SOA Design, Modelling and Analysis
Furthermore, as the wavelength is further lowered from the peak, the measurement is dominated by the noise floor and sensitivity of the system, due to the limited amount of spectral power from the broadband ASE source. This causes the observed inversion in the expected gain drop at short wavelengths. The SOA gain as a function of input power was also investigated at a wavelength of 1550. The small-signal gain of the SOA device is measured to be 15.25 dB. The gain reduction characteristic as a function of input power due to carrier depletion in the active region has also been measured, providing a saturation input power \(P_{in}^{sat}\), at which the gain is reduced by 3 dB, of 5.2 dBm. Signal distortion will occur when the output optical power becomes comparable to the saturation power of the amplifier \((P_{sat})\)[35; 36], defined by:  
\[P_{sat}=\frac{h\delta\;\sigma_{m}}{\Gamma\tau_{c}a} \tag{3}\]  
where \(h\) is the Plank constant, \(\
u\) is the optical frequency, \(\sigma_{m}=W.d\) is the active region cross section, where \(W\) and \(d\) are its width and thickness, respectively, \(\Gamma\) is the optical confinement factor, \(\tau_{c}\) is the carrier life time and \(a\) is the already introduced differential gain coefficient. By using the following equation, the SOA gain as a function of input power was obtained [37]:  
\[G=G_{0}\cdot\exp\left((1-G)\frac{P_{in}}{P_{sat}}\right) \tag{4}\]  
where \(G_{0}\) is the (small-signal) unsaturated amplifier gain, and \(P_{in}\) is the input amplifier power, which can be defined as a function of output power \(P_{in}=P_{out}/G\) where \(P_{out}\) is the output amplifier power. Figure 4 shows the simulated and the experimentally measured gain saturation characteristics for a wavelength of 1550 nm at an injection current of 500 mA. The simulation results are in good agreement with the experimentally measured gain values. It should be noted that the device used in the experiment was a relatively dated one. Current SOA technology is able to provide similar values of small signal gain and output saturation power with driving current in the order of 200 mA. Figure 3: Simulated (blue) and experimental (red) SOA gain profile. The dynamic model used to simulate the NTW SOA is similar to the model used in [38], which is based on the numerical solution of the rate equation for carrier density. The model can be easily updated to include nonlinear gain compression terms [15] to investigate operation under strong saturation regimes as well. For the modal gain, a fitted parabolic model was used:  
\[g=a(n-n_{0})\left(1-\left[\frac{\frac{c}{\lambda}-\frac{c}{x_{opt}}}{c_{v}[n-n_ {0}]}-1\right]^{2}\right) \tag{5}\]  
where \(c_{v}\) is a parameter related to the bandwidth of the modal gain, and \(\lambda_{gap}\) is the band-gap optical wavelength. To perform dynamic simulations, the SOA was divided into a sufficiently large number of sections, so that the carrier concentration density within each section can be considered constant. Starting from the first section, where the input signal is considered, the solution of the time-dependent carrier rate equation for each section is used to calculate the signal at the input of the subsequent section. To solve the rate equations of each section, a standard ordinary differential equation (ODE) is used. The SOA model parameters used for simulations are presented in Table 1.  
In the present paper, the effect of the amplified spontaneous emission to the SOA output signal was added as Gaussian noise with variance equal to the ASE power. The spontaneous emission power can be written as [39]:  
\[P_{ASE}=2h\beta n_{sp}(G-1)BW_{o} \tag{6}\]