We further comment is that if we

We noticed that the dependence is not linear and
becomes increasingly quadratic for a higher concentration. Indeed, equation 1
shows that P22 is determined by a,
and has a quadratic dependence on N2 while the dependence on the
pump power is linear, so the equation balance may justify such a quadratic
dependence. A further comment is that if we underestimate the value of a by a
factor of 3 we significantly underestimate the required intensity (Albalawi, Varas, Chiasera, Gebavi,
Albalawi, et al. 2017).

Finally, we noticed that a low value of a
may be of a higher benefit in the case of a higher inversion, like in the case
of amplifiers. This may suggest that a choice of the most appropriate glass may
depend on the specific application and should take into account the whole set
of parameters, including reverse cross-relaxation. From Figure 2 we can also
see that impact of the reverse cross-relaxation parameter is larger when high
inversion are required, such as in amplifier devices, this may suggest that the
choice of the most effective glass may depend on the specific application. For
example for in Ref. (Jackson and King 1999) the silica glass shows a longer (30%) 3H4
lifetime at around 5.75 mol% doping level, but a = 0.08. This may indicate that the choice of glass may depend on
the specific application. Other glasses shows even lower a, Ref. (Fagundes-Peters, De Camargo, and Nunes 2006) calculated, using Koshida model (Kushida 1973), a
parameter below 0.01 for fluoroindogallate glass. This indicate that a
comparable measurement of reverse cross-relaxation parameter across different
types of glasses may be useful for laser scientist to choose the appropriate
glass host for each specific application.

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