# Neuro-Inspired Computing with Spin-VCSELs
## 2 The Theoretical Model
### The RC Setup  
A schematic overview of the simulated theoretical model is shown on the left in Figure 1. A spin-VCSEL is connected to a delay line, which has a round-trip time \(\tau_{D}\). Data are injected in the VCSEL through the optical spin injection, such that we achieve the following pump rates:  
\[J^{+}=\frac{J_{0}}{2}+\delta JA(t)\quad\text{and}\quad J^{-}=\frac{J_{0}}{2}, \tag{4}\]  
where \(J_{0}\) is the previously mentioned electrical pump, \(\delta J\) is the amplitude of the optical spin injection and \(A(t)\) is the normalized masked data that is to be processed. The masked data \(A(t)\) are obtained via the product \(I(t)M(t)\), where \(I(t)\) is the input data and \(M(t)\) is a sequence periodically repeating a mask. The masked data \(A(t)\) are normalized between 0 and 1. The mask consists of \(N\) step-wise constant levels of duration \(\theta\). Each datapoint in \(I(t)\) is held constant for a duration \(\tau_{M}=N\theta\) that corresponds to the mask length, as shown on the right in Figure 1. The purpose of the different levels in the mask is to create diversity in the nonlinear response of the VCSEL, such that we obtain a diverse set of virtual nodes spread along the delay line. The mask length \(\tau_{M}\) is often matched to the delay time \(\tau_{D}\), however, in this paper, this is not the case and \(\tau_{M}\) can be much larger than \(\tau_{D}\), such that a single masked datapoint is spread over multiple roundtrips of the delay line. A principal mask is generated for a maximum of 100 nodes by randomly selecting each mask level from the following set, \((0,0.25,0.5,0.75,1)\). If \(N<100\), we used the first \(N\) values from this principal mask to obtain our mask. The principal mask is kept fixed. We only inject data in the spin-down carrier population, which contributes to the left circularly polarized mode. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Linewidth enhancement factor & \(\alpha\) & 5 \\ Carrier decay rate & \(\gamma\) & \(1\,\mathrm{ns^{-1}}\) \\ Photon lifetime & \(\tau_{P}\) & \(1.54\,\mathrm{ps}\) \\ Spin decay rate & \(\gamma_{s}\) & \(450\,\mathrm{ns^{-1}}\) \\ Linear dichroism & \(\gamma_{a}\) & \(-1.16\,\mathrm{ns^{-1}}\) \\ Linear birefringence & \(\gamma_{P}\) & \(200\pi\,\mathrm{GHz}\) \\ Amplitude saturation factor & \(\epsilon_{a}\) & \(1.6\) \\ Phase saturation factor & \(\epsilon_{P}\) & \(-3.2\) \\ Electrical pump & \(J_{0}\) & \(2\)\(l_{th}\), unless mentioned otherwise. \\ Optical pump amplitude & \(\delta J\) & \(6\)\(l_{th}\), unless mentioned otherwise. \\ Constant feedback phase & \(\Omega\) & \(0\) \\ Mask length & \(\tau_{M}\) & \(\tau_{M}=N\theta\) \\ Delay time & \(\tau_{D}\) & scanned from 2.5 ps to 1 ns \\ Feedback rate & \(\eta\) & scanned from 1 to 100 ns\({}^{-1}\) \\ Number of nodes & \(N\) & scanned from 5 to 100 \\ Node spacing & \(\theta\) & scanned from 0.5 to 10 ps \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of parameters, designation and values used in the simulations for delay-based RC using the SFM. We do not have fixed values for the node separation \(\theta\), the number of nodes \(N\), nor the delay time \(\tau_{D}\) or mask length \(\tau_{M}\), because this is the first time that delay-based RC using spin-VCSELs is being investigated and we will scan these parameters for optimal RC performance. Typically, \(\theta\) is closely related to the fastest timescale of the laser-based reservoir, which in this case is the polarization oscillation, which on its turn is linearly dependent on the inverse of the birefringence \(\gamma_{p}\). Hence, we expected \(\theta\) to be in the range of 1-10 ps, which is a substantial improvement in comparison with node spacings of 20 ps used in other laser-based RC systems [19]. There are several ways in which the output layer of the RC can be constructed.