To understand the concept of a population inversion, it is necessary to understand the way that light interacts with matter. To do so, it is useful to consider a very simple system of atoms forming a laser medium.
Assume there are a group of N atoms, each of which is capable of being in one of two energy states, either:
The number of these atoms which are in the ground state is given by N1, and the number in the excited state N2. Since there are N atoms in total, N1 + N2 = N .
The energy difference between the two states, ΔE = E2-E1 determines the characteristic frequency ν of light which will interact with the atoms; This is given by Bohr's frequency relation:
E2-E1 = ΔE = hν ,
h being [Planck's constant]?.
If the group of atoms are in [thermal equilibrium]?, it can be shown from thermodynamics that the ratio of the number of atoms in each state is given by a Boltzmann distribution:
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N2 / N1 = exp{ -(E2-E1) / kT } ,
where T is the temperature of the group of atoms, and k is Boltzmann's constant.
We may calculate the ratio of the populations of the two states at room temperature (T≈300 K) for an energy difference ΔE that corresponds to light of a frequency corresponding to visible light (ν≈5*1014 Hz). Since E2 - E1 >> kT, if follows that the argument of the exponential in the equation above is a large negative number, and as such N2 / N1 is vanishingly small, i.e., that there are almost no atoms in the excited state.
When in thermal equilibrium, then, it is seen that the lower energy state is usually more populated than the upper energy state, and this is the normal state of the system. If the ratio could be inverted such that N2/N1 > 1, then the system is said to have a population inversion. In fact, Boltzmann statistics predicts that, for all positive values of ΔE and temperature, the population of N1 will always exceed that of N2, when the system is a thermal equilibrium. It is clear then, that to produce a population inversion, the system cannot be at thermal equilibrium.
There are three types of possible interactions between the system of atom and light that are of interest:
1. Absorption.
If light (i.e., photons) of frequency ν 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 density of the light, and also to the number of atoms currently in the ground state, N1.
If an atom is in the excited state, it will spontaneously decay to the ground state at a rate proportional to N2, the number of atoms in the excited state. The energy difference between the two states ΔE is emitted from the atom as a photon of frequency ν as given by Bohr's relation above.
The photons are emitted stochastic?ally, 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:
N2(t) = N2(0) exp{ -t / τ ),
where N2(0) is the number of excited atoms at time t=0, and τ is the lifetime of the transition between the two states.
If an atom is already in the excited state, it may be perturbed by the passage of a photon which has a frequency ν corresponding to the energy gap ΔE 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. 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 N2 in the excited state, and the radiation density of the light.
The critical detail of stimulated emission is that the induced photon has the same frequency and phase as the inducing 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 happening. Initially, atoms are energised 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 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 affect, 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 rate of these two processes are both proportional to the number of atoms in the ground and excited states, N1 and N2, respectively. 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 N2 / N1 > 1; In other words, a population inversion is required for laser operation.
(more here to come)