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from the ground state to the excited state by a process called pumping , described below. Some of these atoms decay via spontaneous emission, releasing incoherent light as photons of frequency, ν. These photons are fed back into the laser medium, usually by an optical resonator. Some of these photons are absorbed by the atoms in the ground state, and the photons are lost to the laser process. However, some photons cause stimulated emission in excited-state atoms, releasing another coherent photon. In effect, this results in optical amplification .

If the number of photons being amplified per unit time is greater than the number of photons being absorbed, then the net result is a continuously increasing number of photons being produced; the laser medium is said to have a gain of greater than unity.

Recall from the descriptions of absorption and stimulated emission above that the rates of these two processes are proportional to the number of atoms in the ground and excited states, N

1 and N

2 , respectively. If the ground state has a higher population than the excited state ( N

1 > N

2 ), the process of absorption dominates and there is a net attenuation of photons. If the populations of the two states are the same ( N

1 = N

2 ), the rate of absorption of light exactly balances the rate of emission; the medium is then said to be optically transparent .

If the higher energy state has a greater population than the lower energy state ( N

1 <

N

2 ), then the emission process dominates, and light in the system undergoes a net increase in intensity. It is thus clear that to produce a faster rate of stimulated emissions than absorptions, it is required that the ratio of the populations of the two states is such that

N

2 / N

1 > 1; In other words, a population inversion is required for laser operation.

Selection rules

Many transitions involving electromagnetic radiation are strictly forbidden under quantum mechanics. The allowed transitions are

described by so-called selection rules, which describe the conditions under which a radiative transition is allowed. For instance, transitions are only allowed if Δ S =0, S being the total spin angular momentum of the system. In real materials other effects, such as interactions with the crystal lattice, intervene to circumvent the formal rules. In these systems the forbidden transitions can occur, but usually at slower rates than allowed transitions. A classic example is phosphorescence where a material has a ground state with S =0, an excited state with

S =0, and an intermediate state with S =1. The transition from the intermediate state to the ground state by emission of light is slow because of the selection rules. Thus emission may continue after the external illumination is removed. In contrast fluorescence in materials is characterized by emission which ceases when the external illumination is removed.

Transitions which do not involve the absorption or emission of radiation are not affected by selection rules. Radiationless transition between levels, such as between the excited S =0 and S =1 states, may proceed quickly enough to siphon off a portion of the

S =0 population before it spontaneously returns to the ground state.

The existence of intermediate states in materials, as we will see, is essential to the technique of optical pumping of lasers.

Creating a population inversion

As described above, a population inversion is required for laser operation, but cannot be achieved in our theoretical group of atoms with two energy-levels when they are in thermal equilibrium. In fact, any method by which the atoms are directly and continuously excited from the ground state to the excited state (such as optical absorption) will eventually reach equilibrium with the de-exciting processes of spontaneous and stimulated emission. At best, an equal population of the two states, N

1 = N

2 = N /2, can be achieved, resulting in optical transparency but no net optical gain.