# Parallel Extreme Learning Machines Based on Frequency Multiplexing
## 2 Experimental System
### Electric Field Description  
Here, we describe the experiment in terms of electric fields propagating in the setup. The description is similar to what was already proposed in our previous work [19], but now accounts for the presence of multiple light sources. Periodic phase modulation at frequency \(\Omega\) acts on a generic radiation \(E(t)\) according to the following formula:  
\[E(t)\to E(t)e^{-im\cos\left(\Omega t\right)}=E(t)\sum\limits_{k=-\infty}^{+ \infty}i^{k}J_{k}(m)e^{-ik\Omega t}, \tag{1}\]  
where \(m\) is the index describing the phase modulation strength. The series expansion of the phase modulation term \(e^{-im\cos\left(\Omega t\right)}\) is known as the Jacobi-Anger expansion and shows how periodic phase modulation results in the creation of a frequency comb. \(J_{k}(m)\) represents the Bessel function of the first kind. Note that the coefficients of the expansion tend toward zero as \(|k|\) increases. We experimentally noticed a slight asymmetry in the combs, which means that the coefficients in Equation (1) may differ from the real ones; since this effect happens in PM\({}_{1}\), where \(m\) is higher, we believe it is due to higher-order effects on phase modulation (see [19] for details). We assume to have \(N_{S}\) monochromatic input sources characterized by constant amplitudes \(E_{j}^{0}\) with \(j\in[1,\ N_{S}]\). Hence, the input radiation is as follows:  
\[E^{\text{source}}(t)=\sum\limits_{j=1}^{N_{S}}E_{j}^{0}e^{-iw_{j}t}\, \tag{2}\]  
where \(\omega_{j}=c/\lambda_{j}\) represent the frequency of the \(j\)-th monochromatic light source. When passing through PM\({}_{1}\), periodic phase modulation is applied, which results in the creation of \(N_{S}\) frequency combs:  
\[E^{\text{comb}}(t)=\sum\limits_{j=1}^{N_{S}}\sum\limits_{k}E_{j,\ k}^{\text{ comb}}e^{-i(\omega_{j}+k\Omega)t}\,\quad E_{j,\ k}^{\text{comb}}=E_{j}^{0}e^{i\phi_{j}^{(1)}}i^{k}J_{k}(m_{1})\, \tag{3}\]  
where the sum over \(k\) should extend to the whole set of integer indexes, as in (1), but in practice (e.g., in simulations), it can be truncated once the coefficients become small enough. \(\phi_{j}^{(1)}\) represents the phase accumulated by the radiation at frequency \(\omega_{j}\) while travelling from the source to PM\({}_{1}\). The programmable spectral filter SF\({}_{1}\) encodes input values on the comb lines, hence generating the input layer of the neural network:  
Figure 2: (**a**) Example of spectral comb recorded at the output of PM\({}_{1}\). For this measurement, a single CW laser source was employed at the wavelength \(\lambda=1554.6\) nm. RF modulation was \(\Omega/2\pi=16.99175\) GHz, corresponding to a difference in wavelength of approximately 137 pm. The resulting comb has approximately 30 usable lines, which correspond to a bandwidth of approximately 4 nm. (**b**) Example of comb line intensities of the hidden layer, as measured by PD\({}_{1}\) and PD\({}_{2}\). The task being solved is the mushroom classification in the single operation mode. Three combs are used at the same time, each one comprising 31 comb lines numbered from 1 to 31. \[E^{\text{in}}(t)=\sum_{j=1}^{N_{S}}\sum_{k}E^{\text{in}}_{j,\,k}e^{-i(\omega_{j}+ k\Omega)t}\,\quad E^{\text{in}}_{j,\,k}=E^{\text{comb}}_{j,\,k}e^{i\phi^{(2)}_{j,\,k}}\sqrt{ F^{\text{in}}_{j,\,k}}\, \tag{4}\]  
where \(\phi^{(2)}_{j,\,k}\) is the phase accumulated by the \(k\)-th line of the \(j\)-th comb when travelling from PM\({}_{1}\) to SF\({}_{1}\), while \(F^{\text{in}}_{j,\,k}\) is the attenuation that SF\({}_{1}\) applies to the \(k\)-th comb line of the \(j\)-th comb.