Editing Population inversion (section)

=== Three-level lasers ===
[[File:Population-inversion-3level.png|frame|right|A three-level laser energy diagram.]]
To achieve lasting non-equilibrium conditions, an indirect method of populating the excited state must be used. To understand how this is done, consider 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 ''E''<sub>1</sub>, ''E''<sub>2</sub>, and ''E''<sub>3</sub>, and populations ''N''<sub>1</sub>, ''N''<sub>2</sub>, and ''N''<sub>3</sub>, respectively.

Assume ''E''<sub>1</sub> &lt; ''E''<sub>2</sub> &lt; ''E''<sub>3</sub>; 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., ''N''<sub>1</sub> ≈ ''N'', {{nowrap|''N''<sub>2</sub> ≈ ''N''<sub>3</sub> ≈ 0}}. If the atoms are subjected to light of a frequency <math>\scriptstyle\nu_{13} \,=\, \frac{1}{h}\left(E_3 - E_1\right)</math>, the process of optical absorption will excite electrons from the ground state to level 3. This process is called ''[[laser pumping|pumping]]'', and does not necessarily 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 {{nowrap|''E''<sub>1</sub> → ''E''<sub>3</sub>}} as the ''pump transition'', which is shown as the arrow marked '''P''' in the diagram on the right.

Upon pumping the medium, an appreciable number of atoms will transition to level 3, such that {{nowrap|''N''<sub>3</sub> &gt; 0}}. To have a medium suitable for laser operation, it is necessary that these excited atoms quickly decay to level 2. The energy released in this transition may be emitted as a photon (spontaneous emission), however in practice the {{nowrap|3 → 2}} transition called the [[Auger effect]] (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 electron in level 2 may decay by spontaneous emission to the ground state, releasing a photon of frequency ''ν''<sub>12</sub> (given by {{nowrap|1=''E''<sub>2</sub> − ''E''<sub>1</sub> = ''hν''<sub>12</sub>}}), which is shown as the transition '''L''', called the ''laser transition'' in the diagram. If the lifetime of this transition, ''τ''<sub>21</sub> is much longer than the lifetime of the radiationless {{nowrap|3 → 2}} transition ''τ''<sub>32</sub> (if {{nowrap|''τ''<sub>21</sub> ≫ ''τ''<sub>32</sub>}}, known as a ''favourable lifetime ratio''), the population of the ''E''<sub>3</sub> will be essentially zero ({{nowrap|''N''<sub>3</sub> ≈ 0}}) and a population of excited state atoms will accumulate in level 2 ({{nowrap|''N''<sub>2</sub> > 0}}). If over half the ''N'' atoms can be accumulated in this state, this will exceed the population of the ground state ''N''<sub>1</sub>. A population inversion (''N''<sub>2</sub> &gt; ''N''<sub>1</sub> ) has thus been achieved between level 1 and 2, and optical amplification at the frequency ''ν''<sub>21</sub> 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 Maiman]] in 1960). A three-level system could also have a radiative transition between level 3 and 2, and a non-radiative transition between 2 and 1. In this case, the pumping requirements are weaker. In practice, most lasers are ''four-level lasers'', described below.