Editing Active laser medium (section)

===Cross-sections===
The simple medium can be characterized with [[cross section (physics)|effective cross-sections]] of [[Absorption (electromagnetic radiation)|absorption]] and [[Emission (electromagnetic radiation)|emission]] at frequencies <math>~\omega_{\rm p}~</math> and <math>~\omega_{\rm s}</math>.
* Have <math>~N~</math> be concentration of active centers in the solid-state lasers.
* Have <math>~N_1~</math> be concentration of active centers in the ground state.
* Have <math>~N_2~</math> be concentration of excited centers.
* Have <math>~N_1+N_2=N</math>.

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

The rate of transitions of an active center from the ground state to the excited state can be expressed like this: <math>~ 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} } ~</math>.

While the rate of transitions back to the ground state can be expressed like: <math>~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}~</math>,
where <math>~\sigma_{\rm as} ~</math> and <math>~\sigma_{\rm ap} ~</math> are [[Absorption cross section|effective cross-sections]] of absorption at the frequencies of the signal and the pump, <math>~\sigma_{\rm es} ~</math> and <math>~\sigma_{\rm ep} ~</math> are the same for stimulated emission, and
<math>~\frac{1}{\tau}~</math> is rate of the spontaneous decay of the upper level.

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

<math>~ 
\frac
{{\rm d}n_2}
{{\rm d}t}
=
W_{\rm u} n_1 -
W_{\rm d} n_2
</math>,

<math>~ 
\frac{{\rm d}n_1}{{\rm d}t}=-W_{\rm u} n_1 + W_{\rm d} n_2
~</math>
However, these equations keep <math>~ n_1+n_2=1 ~</math>.

The absorption <math>~  A ~</math> at the pump frequency and the gain <math>~  G ~</math> at the signal frequency can be written 
as follows:

<math>~  A = N_1\sigma_{\rm pa} -N_2\sigma_{\rm pe} ~</math> and
<math>~  G = N_2\sigma_{\rm se} -N_1\sigma_{\rm sa} ~</math>.

===Steady-state solution===
In many cases the gain medium works in a continuous-wave or [[quasi-continuous function|quasi-continuous]] regime, causing the time [[derivative]]s of populations to be negligible.

The steady-state solution can be written:

<math>~ n_2=\frac{W_{\rm u}}{W_{\rm u}+W_{\rm d}} ~</math>, 
<math>~ n_1=\frac{W_{\rm d}}{W_{\rm u}+W_{\rm d}}.</math>

The dynamic saturation intensities can be defined:

<math>~ I_{\rm po}=\frac{\hbar \omega_{\rm p}}{(\sigma_{\rm ap}+\sigma_{\rm ep})\tau} ~</math>,
<math>~ I_{\rm so}=\frac{\hbar \omega_{\rm s}}{(\sigma_{\rm as}+\sigma_{\rm es})\tau} ~</math>.

The absorption at strong signal:
<math>~ A_0=\frac{ND}{\sigma_{\rm as}+\sigma_{\rm es}}~</math>.

The gain at strong pump:
<math>~ G_0=\frac{ND}{\sigma_{\rm ap}+\sigma_{\rm ep}}~</math>,
where <math>~ D=
\sigma_{\rm pa}
\sigma_{\rm se}
-
\sigma_{\rm pe}
\sigma_{\rm sa}
~</math>
is determinant of cross-section.

Gain never exceeds value <math>~G_0~</math>, and absorption never exceeds value <math>~A_0  U~</math>.

At given intensities <math>~I_{\rm p}~</math>,  <math>~I_{\rm s}~</math> of pump and signal, the gain and absorption
can be expressed as follows:

<math>~A=A_0\frac{U+s}{1+p+s}~</math>,
<math>~G=G_0\frac{p-V}{1+p+s}~</math>,

where 
<math>~p=I_{\rm p}/I_{\rm po}~</math>, 
<math>~s=I_{\rm s}/I_{\rm so}~</math>, 
<math>~U=\frac{(\sigma_{\rm as}+\sigma_{\rm es})\sigma_{\rm ap}}{D}~</math>, 
<math>~V=\frac{(\sigma_{\rm ap}+\sigma_{\rm ep})\sigma_{\rm as}}{D}~</math> .