# Spectral Separation of Up-Conversion Luminescence Processes
## 1 Introduction  
Separation of complex emission spectra into individual components, corresponding to different emission processes, is of utmost importance for analyzing the emitting media. It had arisen at the very early age of spectroscopy, and continues to be relevant up to this day. The nature of the problem is simple: the registered signal at a given frequency is a sum of signals, corresponding to different emission lines or bands. It is not possible to distinguish components of the emission signal, corresponding to different processes, without using some additional information about these processes. The whole variety of existing and widely applied spectra separation methods can be divided into two major branches: one either fits the spectrum with known shapes, or exploits a chemometry-like approach, changing the respective contribution of different processes and measuring the total signal for each setting. The first approach is extensively discussed and used. Powerful numerical methods have been developed to deal with noise and inference of large number of spectral components. Nowadays, commercial (such as PeakFit by Systat Software Inc. US or Peak Resolve by OMNIC, US) and even free software packages are available, allowing us to perform fitting with a plethora of basic shapes (Gaussian, Lorentzian, etc.) [1, 2]. The first approach is very versatile, and continues to be actively discussed and improved. One can infer spectral derivatives of several orders [3], or use the Fourier transform of the compound spectrum to provide more info for the fitting procedure [4]. Statistical inference techniques, such as maximum likelihood [5] and Bayesian estimation [6, 7], have been applied for the task. The recent vogue of deep learning also found its way to spectral separation [8]. The second approach also boasts quite a long history. It was pioneered by [PERSON] and [PERSON] [9], and [PERSON] and [PERSON] [10]. The second approach does not require any a priori knowledge of component spectra. Registering compound spectra for a sufficient number of different weight settings allows one inferring all the components. The works [9, 10] also birthel an extensive and already well matured field of research. In particular, a plethora of chemometric methods has stemmed from it [11, 12]. The most common way to perform spectral separation by the second approach is to change respective quantities of the emitting agents; for example, the concentrations of different chemicals in the mixture. To our knowledge, not much attention was paid to the possibility of distinguishing spectral components by different effective nonlinearities of the incoherent emission processes associated with these components (here we can only mention a recent work [13]). Separating the incoherent nonlinear processes by pump power dependence means that the spectral components inference can be carried on without a priori assumptions and/or some preliminary modeling with just a single sample by pump power variation. This possibility can be quite useful when researching nonlinear media aiming for properties necessary for particular applications in sensing/imaging, multi-photon absorption, metrology, etc. Here, we present such an approach to spectral separation, and apply it to a particular kind of highly nonlinear media, which commonly demonstrates several nonlinear processes occurring simultaneously [14]. These media are the up-conversing (UC) ones, transforming longer wavelength pump into shorter wavelength luminescence; remarkably, UC processes might not require high pump power to occur. Spectral components of such up-conversion luminescence (UCL) processes are known to exhibit quite versatile dependencies on the pump power [15; 16]. Also, spectral overlapping of UC processes is quite common for media with rather different nature of nonlinearity [17; 18; 19; 20; 21; 22; 23; 24; 25] UCL processes are widely implemented in modern optical and biomedical technologies.