Editing Stimulated emission (section)

=== General gain equation ===
The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensity ''I'' as a function of position ''z'' in the [[gain medium]]:
: <math>{ dI \over dz} = { \gamma_0(\nu)  \over 1 + \bar{g}(\nu) { I(z) \over I_S } }  \cdot I(z) </math>
where <math>I_S</math> is saturation intensity.  To solve, we first rearrange the equation in order to separate the variables, intensity ''I'' and position ''z'':
: <math>{ dI \over I(z)} \left[ 1 + \bar{g}(\nu) { I(z) \over I_S } \right]  =  \gamma_0(\nu)\cdot dz </math>

Integrating both sides, we obtain
: <math>\ln \left( { I(z) \over I_{in} } \right) + \bar{g}(\nu) { I(z) - I_{in} \over I_S} = \gamma_0(\nu) \cdot z</math>
or
: <math>\ln \left( { I(z) \over I_{in} } \right) + \bar{g}(\nu)  {  I_{in} \over  I_S  }  \left( { I(z) \over I_{in}  } - 1 \right)  = \gamma_0(\nu) \cdot z</math>

The gain ''G'' of the amplifier is defined as the optical intensity ''I'' at position ''z'' divided by the input intensity:
: <math>G = G(z) = { I(z) \over I_{in}  } </math>

Substituting this definition into the prior equation, we find the '''general gain equation''':
: <math>\ln \left( G \right) + \bar{g}(\nu)  {  I_{in} \over  I_S  }  \left( G - 1 \right)  = \gamma_0(\nu) \cdot z</math>