Active laser medium

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 that thermal 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 are metastable. 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 of amplification of optical signals, the lasing frequency is called signal frequency. If the externally provided energy required for the signal's amplification is optical, it would necessarily be at the same or higher pump frequency.

The simple medium can be characterized with effective cross-sections of absorption and emission at frequencies ${\displaystyle ~\omega _{\rm {p}}~}$ and ${\displaystyle ~\omega _{\rm {s}}}$.

• 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: ${\displaystyle ~W_{\rm {u}}={\frac {I_{\rm {p}}\sigma _{\rm {ap}}}{\hbar \omega _{\rm {p}}}}+{\frac {I_{\rm {s}}\sigma _{\rm {as}}}{\hbar \omega _{\rm {s}}}}~}$.

While the rate of transitions back to the ground state can be expressed like: ${\displaystyle ~W_{\rm {d}}={\frac {I_{\rm {p}}\sigma _{\rm {ep}}}{\hbar \omega _{\rm {p}}}}+{\frac {I_{\rm {s}}\sigma _{\rm {es}}}{\hbar \omega _{\rm {s}}}}+{\frac {1}{\tau }}~}$, where ${\displaystyle ~\sigma _{\rm {as}}~}$ and ${\displaystyle ~\sigma _{\rm {ap}}~}$ are effective cross-sections of absorption at the frequencies of the signal and the pump, ${\displaystyle ~\sigma _{\rm {es}}~}$ and ${\displaystyle ~\sigma _{\rm {ep}}~}$ 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:

${\displaystyle ~{\frac {{\rm {d}}n_{2}}{{\rm {d}}t}}=W_{\rm {u}}n_{1}-W_{\rm {d}}n_{2}}$,

${\displaystyle ~{\frac {{\rm {d}}n_{1}}{{\rm {d}}t}}=-W_{\rm {u}}n_{1}+W_{\rm {d}}n_{2}~}$ 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:

${\displaystyle ~A=N_{1}\sigma _{\rm {pa}}-N_{2}\sigma _{\rm {pe}}~}$ and ${\displaystyle ~G=N_{2}\sigma _{\rm {se}}-N_{1}\sigma _{\rm {sa}}~}$.

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:

${\displaystyle ~n_{2}={\frac {W_{\rm {u}}}{W_{\rm {u}}+W_{\rm {d}}}}~}$, ${\displaystyle ~n_{1}={\frac {W_{\rm {d}}}{W_{\rm {u}}+W_{\rm {d}}}}.}$

The dynamic saturation intensities can be defined:

${\displaystyle ~I_{\rm {po}}={\frac {\hbar \omega _{\rm {p}}}{(\sigma _{\rm {ap}}+\sigma _{\rm {ep}})\tau }}~}$, ${\displaystyle ~I_{\rm {so}}={\frac {\hbar \omega _{\rm {s}}}{(\sigma _{\rm {as}}+\sigma _{\rm {es}})\tau }}~}$.

The absorption at strong signal: ${\displaystyle ~A_{0}={\frac {ND}{\sigma _{\rm {as}}+\sigma _{\rm {es}}}}~}$.

The gain at strong pump: ${\displaystyle ~G_{0}={\frac {ND}{\sigma _{\rm {ap}}+\sigma _{\rm {ep}}}}~}$, where ${\displaystyle ~D=\sigma _{\rm {pa}}\sigma _{\rm {se}}-\sigma _{\rm {pe}}\sigma _{\rm {sa}}~}$ is determinant of cross-section.

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

At given intensities ${\displaystyle ~I_{\rm {p}}~}$, ${\displaystyle ~I_{\rm {s}}~}$ of pump and signal, the gain and absorption can 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 ${\displaystyle ~p=I_{\rm {p}}/I_{\rm {po}}~}$, ${\displaystyle ~s=I_{\rm {s}}/I_{\rm {so}}~}$, ${\displaystyle ~U={\frac {(\sigma _{\rm {as}}+\sigma _{\rm {es}})\sigma _{\rm {ap}}}{D}}~}$, ${\displaystyle ~V={\frac {(\sigma _{\rm {ap}}+\sigma _{\rm {ep}})\sigma _{\rm {as}}}{D}}~}$ .

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 a gain medium can be defined as ${\displaystyle ~E={\frac {I_{\rm {s}}G}{I_{\rm {p}}A}}~}$.

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

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.

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