 Solution

In the ideal three-level laser, pumping excites atoms out of the ground state to a higher auxiliary level, rapid relaxation fills a relatively long-lived upper laser level, and the laser transition returns to the ground state. A ruby laser is a standard example of this classification. Since the lower laser state initially holds most atoms, substantial pumping is needed to reverse the population per sublevel. With upper and lower degeneracies $g_2,g_1$ and $N_1+N_2\simeq N_T$, the positive-gain condition isFor equal degeneracies more than half the active atoms must be excited before optical gain is even positive.

In a four-level laser the lower laser level is above the ground reservoir and empties rapidly after laser emission. Pumping again feeds a higher auxiliary level which relaxes into the upper laser state, but the lower laser-state population can remain small. The helium-neon laser provides a multilevel physical realization often represented by this scheme for a four-level laser. Positive population inversion then requires much less depletion of the ground reservoir, giving a much lower laser pump threshold for comparable lifetimes, emission cross sections and cavity losses.

The threshold of a four-level laser is not literally zero. Oscillation requires round-trip optical gain to overcome output coupling and other losses, and competing decay or incomplete lower-level removal raises the required pump. Real laser media contain more atomic states than ideal diagrams for a three-level laser or a four-level laser; the labels describe the relevant pumping and laser-transition cycle rather than the total number of states in the material.