We Table 1. The pump cross section at

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We used the energy levels
scheme and considered transitions shown in Fig. 1(Evans et al. 2009, Walsh et al. 2004).

Fig. 1: Energy level scheme of Thulium

Fig.1 shows the lowest
four energy manifolds of Tm3+ ion. In the figure, the laser
transition, the pump transition, and direct and reverse cross-relaxation
processes are indicated, together with spontaneous decay paths. The
corresponding set of rate equations is as follows:

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where N1 ,N2 ,N3 and
N4 are the population of the energy levels 3H6
(ground level), 3F4 (upper laser level), 3H5
and 3H4 (pump level), 
respectively; W14 ,W41 are  the pump rates,  the lifetime of the i-level, and  are branch ratios from the – to -level (Gebavi et al. 2010). The
coefficients  describe the energy transfer
processes: P41 (3H4, 3H63F4, 3F4)
is the cross relaxation constant, which is proportional to doping level (Gebavi et al. 2010, Taher et al. 2011), and
P22 (3F4,3F4,3H6, 3H4)
is the reverse cross-relaxation process constant, the investigation of which is
the main aim of this study.

1.   Simulation

We use the model to calculate and assesst the reverse
cross relaxation of a set on Tellurite glass samples with doping levels ranging
from 0.36 mol% to 10 mol%. Used numerical values are listed in Table 1. The
pump cross section at 790 nm was taken from Ref (Gebavi et al. 2010) while
the emission cross-section was calculated using Ref (McCumber 1964).

Table 1:List of parameters used
in the modeling

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