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# Active laser medium

Within a laser, the active laser medium or gain medium is the material that exhibits optical gain. This gain is generally generated by stimulated emission on electronic or molecular transitions to a lower energy state, starting from a higher energy state to which it had been previously stimulated by means of a pump source.

Examples of active laser media include:

Pumping of gain media (i.e., the supply of energy) can be achieved with electrical currents (e.g. in semiconductors, or in gases via high-voltage discharges) or with light, which may be generated with discharge lamps or with other lasers (often semiconductor lasers, see DPSS laser). More exotic gain media can be pumped by chemical reactions (see chemical laser), nuclear fission (see nuclear pumped laser), or with high-energy electron beams.[1].

## Example of a model of gain medium

There is no universal model, which would be valid for all types of lasers [2]: .

The simplest model includes two systems of sublevels, upper (2) and lower (1). Within each level the fast transitions lead to the Boltzman distribution of excitations among sublevels (fig.1). The upper level is assumed to be metastable. In this approximation, neither gain, nor refractive index depend on the particular way of excitation. For good performance of the gain medium, the separation between sublevels should be larger than working temperature, then, at the pump frequency $~\omega_{\rm p}~$, the absorption dominates, and the emission (preferably, stimulated emission is dominant at the laser frequency $~\omega_{\rm s}~$. In the case of amplification of optical signals, the lasing frequency is called signal frequency. However, the same term is used even in the laser oscillators, when the amplified radiation is used to transfer energy rather than some information.

The model below seems to work well for most of optically-pumped solid-state lasers. For other types of lasers, (for example, chemical lasers or gas-dynamical lasers) more complicated analysis is required.

#### Cross-sections

The simple medium can be characterized with effective cross-sections of absorption and emission at frequencies $~\omega_{\rm p}~$ and $~\omega_{\rm s}~$. Let $~N~$ be concentration of active centers. This is typical state in the solid-state lasers. Let $~N_1~$ be concentration of active centers in the ground state and Let $~N_2~$ be concentration of excited centers; Let $~N_1+N_2=N~$.

The relative concentrations can be defined as $~n_1=N_1/N~$ and $~n_2=N_2/N~$.

The rate of transitions of an active center from ground state to the excited state can be expressed with $~ W_{\rm u}= \frac{I_{\rm p}\sigma_{\rm ap}}{ \hbar \omega_{\rm p} } +\frac{I_{\rm s}\sigma_{\rm as}}{ \hbar \omega_{\rm s} } ~$ and The rate of transitions back to the ground state can be expressed with $~ W_{\rm d}=\frac{ I_{\rm p} \sigma_{\rm as}}{ \hbar \omega_{\rm p} } +\frac{I_{\rm s}\sigma_{\rm es}}{ \hbar \omega_{\rm s} } +\frac{1}{\tau}~$, where $~\sigma_{\rm as} ~$ and $~\sigma_{\rm ap} ~$ are effective cross-sections of the absorption at the frequencies of the pump and the signal, and $~\sigma_{\rm es} ~$ and $~\sigma_{\rm ep} ~$ are the same for stimulated emission; $~\frac{1}{\tau}~$ is rate of the spontaneous decay of the upper level.

Then, the kinetic equation for relative populations can be written as follows:

$~ \frac {{\rm d}n_2} {{\rm d}t} = W_{\rm u} n_1 - W_{\rm d} n_2 ~$,

$~ \frac{{\rm d}n_1}{{\rm d}t}=-W_{\rm u} n_1 + W_{\rm d} n_2 ~$

However, these equations keep $~ n_1+n_2=1 ~$.

The absorption $~ A ~$ at the pump frequency and the gain $~ G ~$ at the signal frequency can be written as follows:

$~ A = N_1\sigma_{\rm pa} -N_2\sigma_{\rm pe} ~$,

$~ G = N_2\sigma_{\rm se} -N_1\sigma_{\rm se} ~$.

In many cases, the gain medium works in continuous-wave or quasi-contiunuous regime, then the time derrivatives of populations are negligible. The steady-state solution can be written

$~ n_2=\frac{W_{\rm u}}{W_{\rm u}+W_{\rm d}} ~$

$~ n_1=\frac{W_{\rm d}}{W_{\rm u}+W_{\rm d}} ~$

The dynamic saturation intensities can be defined with

$~ I_{\rm po}=\frac{\hbar \omega_{\rm p}}{(\sigma_{\rm ap}+\sigma_{\rm ep})\tau} ~$,

$~ I_{\rm so}=\frac{\hbar \omega_{\rm s}}{(\sigma_{\rm as}+\sigma_{\rm es})\tau} ~$. _ The absorption at strong signal $~ A_0=\frac{ND}{\sigma_{\rm as}+\sigma_{\rm es}}~$.

The gain at strong pump $~ G_0=\frac{ND}{\sigma_{\rm ap}+\sigma_{\rm ep}}~$,

where $~ D= \sigma_{\rm pa} \sigma_{\rm se} - \sigma_{\rm pe} \sigma_{\rm sa} ~$ is determinant of cross-section. For the efficient gain medium, this determinant should be pretty positive.

Gain never exceeds value $~G_0~$, and absorption never exceeds value $~A_0~$.

At given intensities $~I_{\rm p}~$, $~I_{\rm s}~$ of pump and sigmal, the gain and absorption can be expressed as follows:

$~A=A_0\frac{U+s}{1+p+s}~$,

$~G=A_0\frac{p-V}{1+p+s}~$,

where

$~p=I_{\rm p}/I_{\rm po}~$, $~s=I_{\rm s}/I_{\rm so}~$

$~U=\frac{(\sigma_{\rm as}+\sigma_{\rm es})\sigma_{\rm ap}}{D}~$

$~V=\frac{(\sigma_{\rm ap}+\sigma_{\rm ep})\sigma_{\rm as}}{D}~$

#### Identities

The following identities [3] take place:

$~U-V=1~$,

$~A/A_0 +G/G_0=1~$.

The state of gain medium can be characterized with a single parameter. Such a parameter can be population of the upper level, gain or absorption; other parameters can be expressed with the relations above.

#### Efficiency of the gain medium

The efficiency of a gain medium can be defined as

$~ E =\frac{I_{\rm s} G}{I_{\rm p}A}~$.

Without loss of laser cavity, this efficiency would be optical to optical efficiency of the laser. Within the same model, the efficiency can be expressed as follows:

$~E =\frac{\omega_{\rm s}}{\omega_{\rm p}} \frac{1-V/p}{1+U/s}~$.

For the efficient operation, both intensities, pump and signal should exceed their saturation intensities; $~\frac{p}{V}\gg 1~$, and $~\frac{s}{U}\gg 1~$.

The estimates above are valid for the medium uniformly filled with pump and signal light. The spatial hole burning may slightly reduce the efficiency, because some region are pumped well, but the pump is not efficiently withdrawn by the signal in the nodes of the interference of counter-propagating waves.