The pumping threshold power of the laser refers to the pumping power when the laser threshold is satisfied. At this time, the loss in the laser resonator is equal to the small-signal gain. Similar threshold powers exist in other light sources, such as Raman lasers and optical parametric oscillators.

Figure 1. Output versus input power in an optically pumped laser. The pump threshold power is 5W and the slope efficiency is 50%. It should be noted that the curve below the pump threshold power is also slightly puffed up due to the effect of amplified spontaneous emission.

For optically pumped lasers, the threshold pump power can be defined as either the input pump power or the absorbed pump power. For applications, the input pump power is more concerned. But for judging the gain efficiency of the gain medium, the absorbed pump power is more useful.
The low pump threshold power can be obtained when the cavity loss of the resonator is low and the gain efficiency is high. High gain efficiencies are typically obtained with small mode field area gain media with high σ-τ product (emission cross section and upper level lifetime product). The σ-τ product is limited by the transmit bandwidth. Therefore, broadband gain media tend to have higher lasing thresholds.
For a simple quadruple laser gain medium, we can calculate the pump threshold power with a formula:

where Irt is the loss in the resonator, hvp is the photon energy of the pump source, A is the beam area in the laser crystal, ηp is the pump efficiency, τ2 is the upper level lifetime, and σem is the size of the emission cross section.
For a given pump power, optimization of the laser output power typically involves a compromise between high slope efficiency and low laser threshold power. In most cases, the pump power in the working state is several times the pump threshold power. The choice of the optimal pump threshold power is one of the parameters of laser design.
The output power versus laser pump power curve is not always as simple as that shown in Figure 1. For example, in lasers with high resonator loss, the threshold pump power is defined by extrapolating the approximate linearity of the curve at high power to zero under the curve.
There are special lasers, such as single-atom lasers, which have no lasing threshold and are therefore called thresholdless lasers.