Theactive laser medium (also called again medium orlasing medium) is the source of opticalgain within alaser. The gain results from thestimulated emission of photons through electronic or molecular transitions to a lower energy state from a higher energy state previously populated by apump source.

Examples of active laser media include:

• Certaincrystals, typically doped withrare-earthions (e.g.neodymium,ytterbium, orerbium) ortransition metal ions (titanium orchromium); most oftenyttrium aluminium garnet (Y₃Al₅O₁₂),yttrium orthovanadate (YVO₄), orsapphire (Al₂O₃);^[1] and not oftencaesium cadmium bromide (CsCdBr₃) (solid-state lasers)
• Glasses, e.g. silicate or phosphate glasses, doped with laser-active ions;^[2]
• Gases, e.g. mixtures ofhelium andneon (HeNe),nitrogen,argon,krypton,carbon monoxide,carbon dioxide, or metal vapors;^[3] (gas lasers)
• Semiconductors, e.g.gallium arsenide (GaAs),indium gallium arsenide (InGaAs), orgallium nitride (GaN).^[4]
• Liquids, in the form of dye solutions as used indye lasers.^[5][6]

In order to fire a laser, the active gain medium must be changed into a state in whichpopulation inversion occurs. The preparation of this state requires an external energy source and is known aslaser pumping. Pumping may be achieved with electrical currents (e.g. semiconductors, or gases viahigh-voltage discharges) or with light, generated bydischarge lamps or by other lasers (semiconductor lasers). More exotic gain media can be pumped bychemical reactions,nuclear fission,^[7] or with high-energyelectron beams.^[8]

The simplest model of optical gain in real systems includes just two, energetically well separated, groups of sub-levels. Within each sub-level group, fast transitions ensure thatthermal equilibrium is reached quickly. Stimulated emissions between upper and lower groups, essential for gain, require the upper levels to be more populated than the corresponding lower ones. This situation is called population-inversion. It is more readily achieved if unstimulated transition rates between the two groups are slow, i.e. the upper levels aremetastable. Population inversions are more easily produced when only the lowest sublevels are occupied, requiring either low temperatures or well energetically split groups.

In the case ofamplification of optical signals, the lasing frequency is calledsignal frequency. If the externally provided energy required for the signal's amplification is optical, it would necessarily be at the same or higherpump frequency.

The simple medium can be characterized witheffective cross-sections ofabsorption andemission at frequencies${\displaystyle {\omega }_{\mathrm{p}}}$ and${\displaystyle {\omega }_{\mathrm{s}}}$.

• Have${\displaystyle N}$ be concentration of active centers in the solid-state lasers.
• Have${\displaystyle N_1}$ be concentration of active centers in the ground state.
• Have${\displaystyle N_2}$ be concentration of excited centers.
• Have${\displaystyle N_1+N_2=N}$.

The relative concentrations can be defined as${\displaystyle n_1=N_1/N}$ and${\displaystyle n_2=N_2/N}$.

The rate of transitions of an active center from the ground state to the excited state can be expressed like this:${\displaystyle W_{\mathrm{u}}=\frac{I_{\mathrm{p}}{\sigma }_{\mathrm{a}\mathrm{p}}}{\hslash {\omega }_{\mathrm{p}}}+\frac{I_{\mathrm{s}}{\sigma }_{\mathrm{a}\mathrm{s}}}{\hslash {\omega }_{\mathrm{s}}}}$.

While the rate of transitions back to the ground state can be expressed like:${\displaystyle W_{\mathrm{d}}=\frac{I_{\mathrm{p}}{\sigma }_{\mathrm{e}\mathrm{p}}}{\hslash {\omega }_{\mathrm{p}}}+\frac{I_{\mathrm{s}}{\sigma }_{\mathrm{e}\mathrm{s}}}{\hslash {\omega }_{\mathrm{s}}}+\frac{1}{\tau }}$,where${\displaystyle {\sigma }_{\mathrm{a}\mathrm{s}}}$ and${\displaystyle {\sigma }_{\mathrm{a}\mathrm{p}}}$ areeffective cross-sections of absorption at the frequencies of the signal and the pump,${\displaystyle {\sigma }_{\mathrm{e}\mathrm{s}}}$ and${\displaystyle {\sigma }_{\mathrm{e}\mathrm{p}}}$ are the same for stimulated emission, and${\displaystyle \frac{1}{\tau }}$ is rate of the spontaneous decay of the upper level.

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

${\displaystyle \frac{\mathrm{d}{n}_2}{\mathrm{d}t}=W_{\mathrm{u}}{n}_1-W_{\mathrm{d}}{n}_2}$,

${\displaystyle \frac{\mathrm{d}{n}_1}{\mathrm{d}t}=-W_{\mathrm{u}}{n}_1+W_{\mathrm{d}}{n}_2}$However, these equations keep${\displaystyle n_1+n_2=1}$.

The absorption${\displaystyle A}$ at the pump frequency and the gain${\displaystyle G}$ at the signal frequency can be written as follows:

${\displaystyle A=N_1{\sigma }_{\mathrm{p}\mathrm{a}}-N_2{\sigma }_{\mathrm{p}\mathrm{e}}}$ and${\displaystyle G=N_2{\sigma }_{\mathrm{s}\mathrm{e}}-N_1{\sigma }_{\mathrm{s}\mathrm{a}}}$.

In many cases the gain medium works in a continuous-wave orquasi-continuous regime, causing the timederivatives of populations to be negligible.

The steady-state solution can be written:

${\displaystyle n_2=\frac{W_{\mathrm{u}}}{W_{\mathrm{u}}+W_{\mathrm{d}}}}$,${\displaystyle n_1=\frac{W_{\mathrm{d}}}{W_{\mathrm{u}}+W_{\mathrm{d}}}.}$

The dynamic saturation intensities can be defined:

${\displaystyle I_{\mathrm{p}\mathrm{o}}=\frac{\hslash {\omega }_{\mathrm{p}}}{({\sigma }_{\mathrm{a}\mathrm{p}}+{\sigma }_{\mathrm{e}\mathrm{p}})\tau }}$,${\displaystyle I_{\mathrm{s}\mathrm{o}}=\frac{\hslash {\omega }_{\mathrm{s}}}{({\sigma }_{\mathrm{a}\mathrm{s}}+{\sigma }_{\mathrm{e}\mathrm{s}})\tau }}$.

The absorption at strong signal:${\displaystyle A_0=\frac{ND}{{\sigma }_{\mathrm{a}\mathrm{s}}+{\sigma }_{\mathrm{e}\mathrm{s}}}}$.

The gain at strong pump:${\displaystyle G_0=\frac{ND}{{\sigma }_{\mathrm{a}\mathrm{p}}+{\sigma }_{\mathrm{e}\mathrm{p}}}}$,where${\displaystyle D={\sigma }_{\mathrm{p}\mathrm{a}}{\sigma }_{\mathrm{s}\mathrm{e}}-{\sigma }_{\mathrm{p}\mathrm{e}}{\sigma }_{\mathrm{s}\mathrm{a}}}$is determinant of cross-section.

Gain never exceeds value${\displaystyle G_0}$, and absorption never exceeds value${\displaystyle A_{0}U}$.

At given intensities${\displaystyle I_{\mathrm{p}}}$,${\displaystyle I_{\mathrm{s}}}$ of pump and signal, the gain and absorptioncan be expressed as follows:

${\displaystyle A=A_0\frac{U+s}{1+p+s}}$,${\displaystyle G=G_0\frac{p-V}{1+p+s}}$,

where${\displaystyle p=I_{\mathrm{p}}/I_{\mathrm{p}\mathrm{o}}}$,${\displaystyle s=I_{\mathrm{s}}/I_{\mathrm{s}\mathrm{o}}}$,${\displaystyle U=\frac{({\sigma }_{\mathrm{a}\mathrm{s}}+{\sigma }_{\mathrm{e}\mathrm{s}}){\sigma }_{\mathrm{a}\mathrm{p}}}{D}}$,${\displaystyle V=\frac{({\sigma }_{\mathrm{a}\mathrm{p}}+{\sigma }_{\mathrm{e}\mathrm{p}}){\sigma }_{\mathrm{a}\mathrm{s}}}{D}}$ .

The following identities^[9] take place:${\displaystyle U-V=1}$,${\displaystyle A/A_0+G/G_0=1.}$

The state of gain medium can be characterized with a single parameter, such as population of the upper level, gain or absorption.

The efficiency of again medium can be defined as${\displaystyle E=\frac{I_{\mathrm{s}}G}{I_{\mathrm{p}}A}}$.

Within the same model, the efficiency can be expressed as follows:${\displaystyle E=\frac{{\omega }_{\mathrm{s}}}{{\omega }_{\mathrm{p}}}\frac{1-V/p}{1+U/s}}$.

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

The estimates above are valid for a medium uniformly filled with pump and signal light.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.

• Population inversion
• Laser construction
• Laser science
• List of laser articles
• List of laser types

• Gain media Encyclopedia of Laser Physics and Technology

• Laser gain media
• Laser science

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