Active laser medium

The active laser medium or gain medium is the material that exhibits optical gain within a laser. The gain is generated by stimulated emission on electronic or molecular transitions to a lower energy state from a higher energy state previously stimulated by a pump source.

Examples of active laser media include:
 * Certain crystals, typically doped with rare-earth ions (e.g. neodymium, ytterbium, or erbium) or transition metal ions (titanium or chromium); most often yttrium aluminium garnet (YAG), yttrium orthovanadate (YVO4), or sapphire (Al2O3);
 * Glasses, e.g. silicate or phosphate glasses, doped with laser-active ions;
 * Gases, e.g. mixtures of helium and neon (HeNe), nitrogen, argon, carbon monoxide, carbon dioxide, or metal vapors;
 * Semiconductors, e.g. gallium arsenide (GaAs), indium gallium arsenide (InGaAs), or gallium nitride (GaN);

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

Example of a model of gain medium
A universal model valid for all laser types does not exist. The simplest model includes two systems of sub-levels: upper and lower. Within each level, the fast transitions lead to the Boltzmann distribution of excitations among sub-levels (fig.1). The upper level is assumed to be metastable, neither gain nor refractive index depend on a particular way of excitation.

For good performance of the gain medium, the separation between sub-levels should be larger than working temperature; then, at pump frequency $$~\omega_{\rm p}~$$, the absorption dominates.

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 amplified radiation is used to transfer energy rather than information. The model below seems to work well for most optically-pumped solid-state lasers.

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 in the solid-state lasers.
 * Let $$~N_1~$$ be concentration of active centers in the ground state.
 * 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 absorption at the frequencies of the pump and the signal.

$$~\sigma_{\rm es} ~$$ 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: $$~ A = N_1\sigma_{\rm pa} -N_2\sigma_{\rm pe} ~$$, $$~ G = N_2\sigma_{\rm se} -N_1\sigma_{\rm se} ~$$.

Steady-state solution
In many cases the gain medium works in a continuous-wave or quasi-continuous regime, causing the time derivatives of populations to be 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: $$~ I_{\rm po}=\frac{\hbar \omega_{\rm p}}{(\sigma_{\rm ap}+\sigma_{\rm ep})\tau} ~$$, $$~ I_{\rm so}=\frac{\hbar \omega_{\rm s}}{(\sigma_{\rm as}+\sigma_{\rm es})\tau} ~$$.

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.

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 signal, the gain and absorption can be expressed as follows: $$~A=A_0\frac{U+s}{1+p+s}~$$, $$~G=G_0\frac{p-V}{1+p+s}~$$,

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 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 as population of the upper level, gain or absorption.

Efficiency of the gain medium
The efficiency of a gain medium can be defined as $$~ E =\frac{I_{\rm s} G}{I_{\rm p}A}~$$.

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

References and notes
Medi actiu Milieu amplificateur Aktyvioji terpė Лазерные материалы