# Spectral Separation of Up-Conversion Luminescence Processes
## 2 Spectrum Separation Method  
Our spectral separation problem can be formulated in the following generic way. The compound wavelength and pump-dependent measured signal \(\mathsf{J}(\lambda,\mathsf{I})\) is represented as a sum of \(N\) components,  
\[\mathsf{J}(\lambda,\mathsf{I})=\sum_{j=1}^{N}\mathsf{J}_{j}(\lambda,\mathsf{I}), \tag{1}\]  
where the function \(\mathsf{J}_{j}(\lambda,\mathsf{I})\) describe a contribution of \(j\)-th process to the registered signal. We assume that finding each \(\mathsf{J}_{j}(\lambda,\mathsf{I})\) constitutes the solution of the spectral separation problem, providing sufficient information about the underlying processes. The key assumption of our separation method is factorization of the spectral and power dependence for each component process  
\[\mathsf{J}_{j}(\lambda,\mathsf{I})=f_{j}(\lambda)g_{j}(\mathsf{I}), \tag{2}\]  
where the power parts \(g_{j}(\mathsf{I})\) are assumed to be known, and the spectral parts \(f_{j}(\lambda)\) are to be found. When, for any given \(\lambda_{I}\), one can make measurements for a set of \(M\geq N\) powers \(\{\mathsf{I}_{k}\}\), satisfying the condition (2), and the rank of the matrix \(g_{j}(\mathsf{I}_{k})\) is \(N\), the solution of the spectral separation problem for the vector \(f_{j}(\lambda_{I})\) is a standard linear inversion with semi-positivity constraints (i.e., \(f_{j}(\lambda)\geq 0\)). One can implement a plethora of numerical methods for it, starting, for example, with linearly constrained least squares estimation [11; 12]. Repeating the procedure for a necessary set of wavelength one, thus would perform the spectral separation. However, practical realization of this seemingly simple procedure faces two major difficulties (here, we are not addressing any possible conditions on the spectral functions \(f_{j}(\lambda)\), making it necessary to perform inference for a set of wavelength values together, and not for each one separately). Firstly, for a given sample of a nonlinear medium to be analyzed, one generally does not know the functions \(g_{j}(\mathsf{I}_{k})\) beforehand. Secondly, fulfillment of the factorization condition (2) should also be verified. Let us demonstrate ways to deal with these problems with an example of up-conversion processes involving rare-earth ions. For them, it is quite common, exhibiting a UCL proportional to  
\[g(\mathsf{I}_{j})\propto\mathsf{I}^{nj}, \tag{3}\]  
where the number \(nj\) is commonly referred as the \"photonity\" of the \(j\)-th process (or, more conveniently, the \"effective nonlinearity\", EN) [15; 16]. Dependence (3) seems intuitive and straightforward. At least when there is no saturation, one expects this EN to correspond to the number of absorbed photons per emitted photon at a given wavelength, just like it is, for example, for frequency doubling by three-wave mixing. However, for complicated multi-level systems, such as rare-earth ions (Figure 1 gives an example of such a system; it will be discussed in more detail in the next section), this simple rule does not generally hold, even if the pump power dependence can be well described by Equation (3). EN might not even be an integer. To add, interaction between ions in the media (for UCL, it is cross-relaxation that can cause such phenomena such as, for example, the \"photon avalanche\" [49; 50; 51; 52]) can break factorization (2), for instance, leading to the component signals of the following form [15; 16]:  
\[\mathsf{J}_{j}(\lambda,\mathsf{I})=\frac{f_{j}(\lambda)\mathsf{I}^{nj}}{1+c_{ j}(\lambda)\mathsf{I}^{kj}}, \tag{4}\]  
where the spectral function \(c_{j}(\lambda)\) and EN \(kj\) describes contribution of the cross-relaxation process to \(j\)-th component signal (such an expression was actually used in the recent work [13] for fitting the component signal). As Equation (4) shows, one might have bothfactorization and power scaling (3), even for larger intensities when the cross-relaxation influence is strong. Thus, we have devised the following practical procedure.