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# Population inversion

In physics, specifically statistical mechanics, the concept of population inversion is of fundamental importance in laser science because the production of a population inversion is a necessary step in the workings of a laser.

## Boltzmann distributions and thermal equilibrium

To understand the concept of a population inversion, it is necessary to understand some thermodynamics and the way that light interacts with matter. To do so, it is useful to consider a very simple assembly 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:

1. The ground state, with energy E1, or,
2. The excited state, with energy E2,
with E2>E1 .

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 ν21 of light which will interact with the atoms; This is given by the relation:

E2-E1 = ΔE = hν21 ,

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:

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.

## The interaction of light with matter

There are three types of possible interactions between the system of atom and light that are of interest:

### Absorption

If light (i.e., photons) of frequency ν21 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.

### Spontaneous emission

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 ν21 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:

N2(t) = N2(0) exp{ -t / τ21 ),

where N2(0) is the number of excited atoms at time t=0, and τ21 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 ν21 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 ν21. 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 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 both proportional to the number of atoms in the ground and excited states, N1 and N2, respectively. If the ground state has a higher population than the excited state (N1 > N2), the process of absorption dominates and there is a net attenuation of photons. If the populations of the two states are the same (i.e., N1 = N2), 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 (N1 < N2), 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 N2 / N1 > 1; In other words, a population inversion is required for laser operation.

### 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, N1 = N2 = N/2, can be achieved, resulting in optical transparency but no net optical gain.

To achieve non-equilibrium conditions, and indirect method of populating the excited state must be used. To understand how this is done, we may use a slightly more realistic model, that of a three-level laser. Again consider a group of N atoms, this time with each atom able to exist in any of three energy states, levels 1, 2 and 3, with energies E1,E2 and E3, and populations N1, N2 and N23, respectively. An energy level diagram of these is shown below:

``` =  level 3, E3, N3
^       |
|       | R (fast, radiationless transition)
|       V
--|--------------------------------  level 2, E2, N2
|                   |
|                   |
| P                 |
| (pump             | L (slow, laser transition)
|  transition)      |
|                   |
|                   V
-----------------------------------  level 1 (ground state), E1, N1
```

Note that E1 < E2 < E3, that is, the energy of level 2 lies between that of the ground state and level 3.

Initially, the system of atoms is at thermal equilibrium, and the majority of the atoms will be in the ground state, i.e. N1N, N2N3 ≈ 0. If we now subject the atoms to light of a frequency ν31, where E3-E1 = hν31 (h being Planck's constant), the process of optical absorption will excite the atoms from the grounds state to level 3. This process is called pumping, and in general does not always directly involve light absorption; other methods of exciting the laser medium, such as electrical discharge or chemical reactions may be used. The level 3 is sometimes referred to as the pump level or pump band, and the energy transition E1 -> E3 as the pump transition, which is shown as the arrow marked P in the diagram above.

If we continue pumping the atoms, we will excite an appreciable number of them into level 3, such that N3 > 0. In a medium suitable for laser operation, we require these excited atoms to quickly decay to level 2. The energy released in this transition may be emitted as a photon (spontaneous emission), however in practice the 3->2 transition (labeled R in the diagram) is usually radiationless, with the energy being transferred to vibrational motion (heat) of the host material surrounding the atoms, without the generation of a photon.

An atom in level 2 may decay by spontaneous emission to the ground state, releasing a photon of frequency ν21 (given by E2-E1 = hν21), which is shown as the transition L, called the laser transition in the diagram. If the lifetime of this transition, τ21 is much longer than the lifetime of the radiationless 3->2 transition τ32, (i.e., if τ21 >> τ32), the population of the E3 will be essentially zero (N3 ≈ 0) and a population of excited state atoms will accumulate in level 2 (N2 > 0). If over half the N atoms can be accumulated in this state, this will exceed the population of the ground state N1. A population inversion (N2 > N2 ) has thus been achieved between level 1 and 2, and optical amplification at the frequency ν21 can be obtained.

Because at least half the population of atoms must be excited from the ground state to obtain a population inversion, the laser medium must be very strongly pumped. This makes three-level lasers rather inefficient, despite being the first type of laser to be discovered (based on a ruby laser medium, by Theodore H. Maiman in 1960). In practice, most lasers are four-level lasers, as shown in the following energy diagram:

``` =  level 4, E4, N4
^       |
|       | Ra (fast, radiationless transition)
|       V
--|--------------------------------  level 3, E3, N3
|                   |
|                   |
| P                 |
| (pump             | L (slow, laser transition)
|  transition)      |
|                   |
|                   V
--|--------------------------------  level 2, E2, N2
|        |
|        | Rb (fast, radiationless transition)
|        V
-----------------------------------  level 1 (ground state), E1, N1
```

Here, there are four energy levels, energies E1, E2, E3, E4, and populations N1, N2, N3, N3, respectively. The energies of each level are such that E1 < E2 < E3 < E4.

In this system, the pumping transition P excites the atoms in the ground state (level 1) into the pump band (level 4). From level 4, the atoms again decay by a fast, non-radiative transition Ra into the level 3. Since the lifetime of the laser transition L is long compared to that of Ra32 >> τ43), a population accumulates in level 3 (the upper laser level), which may relax by spontaneous or stimulated emission into level 2 (the lower laser level). This level likewise has a fast, non-radiative decay Rb into the ground state.

As before, the presence of a fast, radiationless decay transitions result in population of the pump band being quickly depleted (N4 ≈ 0). In a four-level system, any atom in the lower laser level E2 is also quickly de-excited, leading to a negligible population in that state (N2 ≈ 0). This is important, since any appreciable population accumulating in level 3, the upper laser level, will form a population inversion with respect to level 2. That is, as long as N3 > 0, then N3 > N2 and a population inversion is achieved. Thus optical amplification, and laser operation, can take place at a frequency of ν32 (E3-E2 = hν32).

Since only a small number of atoms must be excited into the upper laser level to form a population inversion, a four-level laser is much more efficient than a three-level one, and most practical lasers are of this type. In reality, many more than four energy levels may be involved in the laser process, with complex excitation and relaxation processes involved between these levels. In particular, the pump band may consist of several distinct energy levels, or a continuum of levels, which allow optical pumping of the medium over a wide range of wavelengths.

Note that in both three- and four-level lasers, the energy of the pumping transition is greater than that of the laser transition. This means that, if the laser is optically pumped, the frequency of the pumping light must be greater than that of the resulting laser light. In other words, the pump wavelength is shorter than the laser wavelength. It is possible in some media to use multiple photon absorptions between multiple lower-energy transitions to reach the pump level; such lasers are called up-conversion lasers.

While in many lasers the laser process involves the transition of atoms between different electronic energy states, as described in the model above, this is not the only mechanism that can result in laser action. For example, there are many common lasers (e.g. dye lasers, carbon dioxide lasers) where the laser medium consists of complete molecules, and energy states correspond to vibrational and rotational modes of oscillation of the molecules. This is the case with water masers, that occur in nature.