## 2 CR-SOA
### Numerical Analysis  
The CR-SOA's dynamic behavior is governed by time-dependent gain equations that account for carrier recombination between the active region (AR) and the carrier  
Figure 1: Carrier transition processes in CR-SOA, illustrating the movement of carriers between the carrier reservoir (CR) and the active region (AR) during operation. reservoir (CR), along with ultrafast intraband effects such as carrier heating (CH) and spectral hole burning (SHB). These effects are crucial in determining the device's overall gain response and high-speed operation. The governing equations can be expressed as follows [11; 12; 15; 18; 21]:  
\[\frac{dh_{AR}(t)}{dt}\ =\ \frac{h_{CR}(t)-h_{AR}(t)}{\tau_{t}(1+\eta)}\ +\ \frac{\eta h_{0}}{\tau_{c}(1+\eta)}\ -\ \frac{h_{AR}(t)}{\tau_{c}}\ -\ ( exp\left[h_{AR}(t)\ +\ h_{CH}(t)\ +\ h_{SHB}(t)\right]\ -\ 1)\ \frac{P_{in,\ CR-SOA}(t)}{E_{sat}} \tag{1}\]  
\[\frac{dh_{CR}(t)}{dt}\ =\ -\frac{\eta(h_{CR}(t)-h_{AR}(t)}{\tau_{t}(1+\eta)}\ +\ \frac{h_{0}-h_{CR}(t)}{\tau_{c}(1+\eta)}\ -\ \frac{h_{CR}(t)}{\tau_{c}} \tag{2}\]  
\[\frac{dh_{CH}(t)}{dt}\ =\ -\frac{h_{CH}(t)}{\tau_{CH}}\ -\ \frac{\varepsilon_{CH}}{ \tau_{CH}}(exp\left[h_{AR}(t)\ +\ h_{CH}(t)\ +\ h_{SHB}(t)\right]\ -\ 1)\,P_{in,\ CR-SOA}(t) \tag{3}\]  
\[\frac{dh_{CH}(t)}{dt}\ =\ -\frac{h_{CH}(t)}{\tau_{CH}}\ -\ \frac{\varepsilon_{CH}}{ \tau_{CH}}(exp\left[h_{AR}(t)\ +\ h_{CH}(t)\ +\ h_{SHB}(t)\right]\ -\ 1)\,P_{in,\ CR-SOA}(t) \tag{4}\]  
where  
* \(\tau_{t}\): Transition lifetime from CR to AR
* \(\tau_{c}\): Carrier lifetime
* \(\eta\): Population inversion factor, defined as the ratio of carrier densities in the active region (AR) and carrier reservoir (CR), i.e., \(\eta=\mathrm{N}_{AR}/\mathrm{N}_{CR}\). * \(\tau_{CH}\): Temperature relaxation rate due to CH
* \(\tau_{SHB}\): Temperature relaxation rate due to SHB
* \(\varepsilon_{CH}\): Nonlinear gain suppression factor due to CH
* \(\varepsilon_{SHB}\): Nonlinear gain suppression factor due to SHB  
The total gain of the CR-SOA is determined by contributions from the AR recombination, CH, and SHB, represented as  
\[G_{CR-SOA_{i}}(t)\ =\ exp\left[h_{AR}(t)\ +\ h_{CH}(t)\ +\ h_{SHB}(t)\right],\ \ \ \ i=1,2,\ldots\ldots. \tag{5}\]  
The unsaturated power gain, G\({}_{0}\), is given by  
\[G_{0}=exp[h_{0}]\ =\ \Gamma\alpha\ \Big{(}\frac{I\tau_{c}}{eV}\ -N_{tr}\Big{)}L \tag{6}\]  
where  
* \(\Gamma\): Optical confinement factor
* \(\alpha\): Differential gain
* \(\mathrm{N}_{tr}\): Transparency carrier density
* I: Injection current
* e: Electron charge
* V = wdL: Active layer volume, where w is the width, d is the thickness, and L is the length of the active region. The saturation energy E\({}_{sat}\) is defined as  
\[E_{sat}=P_{sat}\ \tau_{c}=\ \frac{wdhv}{\Gamma\alpha} \tag{7}\]  
where  
* \(\mathrm{P}_{sat}\): Saturation power
* h: [PERSON]'s constant
* \(\upsilon\): Signal frequency For numerical simulations, the input optical power pulses \(\rm P_{in,CR-SOA}(t)\) are modeled as Gaussian-shaped pulses within a pseudorandom binary sequence (PRBS) of length N, with energy \(\rm E_{0}\), full-width at half maximum (FWHM) \(\rm\tau_{FWHM}\), and bit period T:  
\[P_{A,B}(t)\ \equiv P_{in,CR-SOA}(t)=\ \sum_{n=1}^{N}a_{n(A,B)}\frac{2\sqrt{ \ln[2]}\ E_{0}}{\sqrt{\pi}\ \rm\tau_{FWHM}}\ \exp\left[-\frac{4\ln[2](t\ -\ nT)^{2}}{\tau_{FWHM}^{2}}\right] \tag{8}\]  
where \(\rm a_{n(A,B)}\) represents the PRBS data streams for inputs A and B, with each taking values of '0' or '1'. The induced phase shift in the CR-SOA is given by  
\[\Phi_{CR-SOA_{i}}(t)\ =\ -\ 0.5\ (\alpha\,h_{AR}(t)\ +\ \alpha_{CH}\,h_{CH}(t)),\ \ \ i=1,2,\ldots \tag{9}\]  
where \(\alpha\) is the traditional linewidth enhancement factor (\(\alpha\)-factor) and \(\alpha_{\rm CH}\) is the linewidth enhancement factor for CH. The SHB contribution to phase shift is negligible, i.e., \(\rm\alpha_{SHB}\approx 0\)[15; 18; 21]. This set of equations provides a comprehensive numerical framework for analyzing the CR-SOA's dynamic gain and phase response under high-speed optical signal modulation.