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T he addition o f sen sitisers chrom ium and thulium to holm ium has allow ed the development o f efficient room temperature holmium lasers. However, the low lying terminal laser level still influences the laser performance. Bowman et al report*^^ that, with changes in the output coupler reflectivity, different laser transitions operate in the Ifxm region, due to the different thermal populations o f the terminal laser levels. Previously, Johnson et aP ‘ had observed output at three different wavelengths which depended only on the operating temperature. Thermal population o f the sublevels o f the terminal laser level also affects the laser threshold.

In the early years o f holmium lasers Remski and Smith^^ demonstrated the effect o f temperature by measuring the threshold pulse energy required to produce lasing in a 50mm long, 3mm<^ afiHo:YAG rod. Below 200K the threshold was found to increase alm ost linearly with temperature at a rate o f about 0 .1 J K’* whereas, above 200K , the rate increased rapidly approaching 1.3 J K*' at room temperature. Lotem et al^^ repeated the work o f Remski and Smith but additionally measured the changes in slope efficien cy with temperature comparing aB H oiY L F to ailHo:YAG. D u e to a peak in the slo p e e ffic ie n c y cu rve o f the Y LF crystal, they found the maximum overall laser efficien cy was at 130K, rather than at the low est operating temperature. However, the YAG sample did not show such a peak and the maximum output was, therefore, obtained for the low est temperatures. The reason for this is unexplained but may be, in part, due to any o f the temperature dependent effects id en tified by Armagan et al. T h ese in clud e temperature induced changes in the fluorescent lifetim e o f the ^H^ thulium level^^, which in YAG has a different form to YLF. Additionally, in YAG, temperature dependencies have been identified in the intensity o f the chromium emission (to transfer excitation energy to thulium) and the lifetim e o f the chromium emission^^. Successful modelling o f the combined thermal effects is still awaited.

Several authors have reported attempts to predict the changes in threshold energy o f the CTH: YAG crystal at room temperatures. The most ambitious, to date, has been the attem pt o f Teichm ann et al^- w ho considered the gain con d ition s at threshold and related these to the thermal populations o f both the Tm ^H^ and Ho lev els using a single experim ental point to deduce the 'excitation' factor, the only unknown in their analysis. The resulting prediction was observed to rapidly diverge from the set o f experimental data points although both theory and data points showed a linear dependence with temperature over the temperature range. It is the aim o f this w ork to d eterm in e w h eth er a sim p ler ap p roach , c o n sid erin g o n ly the therm al

population o f the Ho level, provides an adequate description o f the variation in threshold energy.

2 .3 .1 Theoretical considerations

Laser action in the holmium ion takes place between and lev els. N ine laser transitions have been identified so far^"^, the strongest transition being at 2.1 2 1/xm. C hanges in therm al population o f the term inal laser le v e l^ \ as w ell as mirror reflectivity^'* can determ ine the w avelen gth at w hich laser operation is ob served . From the work o f Bowman et al^"* the four strongest transitions have terminal laser levels lying between 492cm * and 522cm * above ground level.

The thermal populations o f two energy levels are related by the Boltzmann equation

N , = N^exp-((E^-E,)/kT) (3)

w here Nj and are the populations at energy levels Ej and E^, where E^ > Ej. H ere, k is B oltzm an n 's constant and T is the absolute tem perature. In norm al, thermal equilibrium conditions, the population difference between two levels E^ and Ej is

= N ,[l-exp (-(E ,-E ,)/k T )] (4)

Siegman*^ develops the theory for population inversion in a four level laser relating the difference between upper and lower laser levels, N^-N^ to the atomic pump rate, Wp, as

N3-N , = J i / L i a w r . , . (5) l + ( l + B)o:WpT^j

where B is the ratio o f populations N^/N^, a is a quantum efficiency depending on the d ecay rates o f the pump le v e l and the upper laser le v e ls and is the radiative lifetim e between the upper and low er laser lev els. is sim ply the sum o f all the populations involved in the laser process. Near threshold B ~ 1 and, if w « 0 .5 , then equation (5) reduces to

where is the atomic pump rate at threshold and is assumed to be proportional to the energy discharged by the flashlamp. Assuming that the population o f at laser threshold is fixed by the thermal conditions and that population o f level does not reduce N^, then equation (6) becomes

N3-N ,exp[-E ,/kT ] = N„(l-B)w^ (7)

where N , is the ground level population. If Nj is constant, again either population o f level N3 does not deplete it, or the level o f depletion is an insignificant amount, and the fluorescent lifetim e is constant over the temperature range^^ then the pump rate required to populate state N^, then becomes temperature sensitive. Assuming is proportional to the flashlamp pump pulse intensity, I.^, via

w = GI,„ (8)

w here G is sim ply a proportionality factor w hich in clu d es the laser pump pulse duration, then equation (7) can be rewritten

E.„^, = (9)

where E.^ is the energy required to be delivered during the pump pulse in order to reach threshold. The threshold energy is therefore expected to vary with temperature according to the thermal population o f the lower laser level.

Siegm an also describes the factors contributing to the output intensity o f a laser as

u = (1 0)

2