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corresponds to light of a frequency corresponding to visible light (ν ≈ 5×10 14 Hz). In this case Δ E = E

2 - E

1 ≈ 2.07 eV, and kT ≈ 0.026 eV. Since E

2 - E

1 kT , it follows that the argument of the exponential in the equation above is a large negative number, and as such N

2 / N

1 is vanishingly small; i.e., there are almost no atoms in the excited state. When in thermal equilibrium, then, it is seen that the lower energy state is more populated than the higher energy state, and this is the normal state of the system. As T increases, the number of electrons in the high-energy state ( N

2 ) increases, but N

2 never exceeds N

1 for a system at thermal equilibrium; rather, at infinite temperature, the populations N

2 and

N

1 become equal. In other words, a population inversion ( N

2 / N

1 > 1) can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state.

The interaction of light with matter

There are three types of possible interactions between a system of atoms and light that are of interest:

Absorption

If light (photons) of frequency ν

12 pass through the group of atoms, there is a possibility of the light being absorbed by atoms which are in the ground state, which will cause them to be excited to the higher energy state. The probability of absorption is proportional to the radiation intensity of the light, and also to the number of atoms currently in the ground state, N

1 .

Spontaneous emission

If a collection of atoms are in the excited state, spontaneous decay events to the ground state will occur at a rate proportional to N

2 , the number of atoms in the excited state. The energy difference between the two states Δ E

21 is emitted from the atom as a photon of frequency ν

21 as given by the frequency-energy relation above.

The photons are emitted stochastically, and there is no fixed phase relationship between photons emitted from a group of excited atoms; in other words, spontaneous emission is incoherent. In the absence of other processes, the number of atoms in the excited state at time t , is given by

,

where N

2 (0) is the number of excited atoms at time t = 0, and τ

21 is the lifetime of the transition between the two states.

Stimulated emission

If an atom is already in the excited state, it may be perturbed by the passage of a photon which has a frequency ν

21 corresponding to the energy gap Δ E of the excited state to ground state transition. In this case, the excited atom relaxes to the ground state, and is induced to produce a second photon of frequency ν

21 . The original photon is not absorbed by the atom, and so the result is two photons of the same frequency. This process is known as stimulated emission . The rate at which stimulated emission occurs is proportional to the number of atoms N

2 in the excited state, and the radiation density of the light. The base probability of a photon causing stimulated emission in a single excited atom was shown by Albert Einstein to be exactly equal to the probability of a photon being absorbed by an atom in the ground state. Therefore, when the numbers of atoms in the ground and excited states are equal, the rate of stimulated emission is equal to the rate of absorption for a given radiation density.

The critical detail of stimulated emission is that the induced photon has the same frequency and phase as the incident photon. In other words, the two photons are coherent. It is this property that allows optical amplification, and the production of a laser system. During the operation of a laser, all three light-matter interactions described above are taking place. Initially, atoms are energised