Editing Stimulated emission (section)

== Optical amplification ==
Stimulated emission can provide a physical mechanism for [[optical amplifier|optical amplification]].  If an external source of energy stimulates more than 50% of the atoms in the ground state to transition into the excited state, then what is called a [[population inversion]] is created.  When light of the appropriate frequency passes through the inverted medium, the photons are either absorbed by the atoms that remain in the ground state or the photons stimulate the excited atoms to emit additional photons of the same frequency, phase, and direction. Since more atoms are in the excited state than in the ground state then an amplification of the input [[irradiance|intensity]] results.

The population inversion, in units of atoms per cubic metre, is
: <math>\Delta N_{21} =  N_2 - {g_2 \over g_1} N_1 </math>
where ''g''<sub>1</sub> and ''g''<sub>2</sub> are the [[degenerate energy level|degeneracies]] of energy levels 1 and 2, respectively.

=== Small signal gain equation ===
The intensity (in [[watt]]s per square metre) of the stimulated emission is governed by the following differential equation:
: <math>{ dI \over dz} = \sigma_{21}(\nu) \cdot \Delta N_{21} \cdot I(z) </math>
as long as the intensity ''I''(''z'') is small enough so that it does not have a significant effect on the magnitude of the population inversion.  Grouping the first two factors together, this equation simplifies as
: <math>{ dI \over dz} = \gamma_0(\nu) \cdot I(z) </math>
where
: <math> \gamma_0(\nu) = \sigma_{21}(\nu) \cdot \Delta N_{21} </math>
is the ''small-signal gain coefficient'' (in units of radians per metre).  We can solve the differential equation using [[separation of variables]]:
: <math>{ dI \over I(z)} = \gamma_0(\nu) \cdot dz </math>

Integrating, we find:
: <math>\ln \left( {I(z) \over I_{in}} \right) =  \gamma_0(\nu) \cdot z </math>
or
: <math>  I(z) = I_{in}e^{\gamma_0(\nu) z} </math>
where
: <math> I_{in} = I(z=0) \, </math> is the optical intensity of the input signal (in watts per square metre).

=== Saturation intensity ===
The saturation intensity ''I''<sub>S</sub> is defined as the input intensity at which the gain of the optical amplifier drops to exactly half of the small-signal gain.  We can compute the saturation intensity as
: <math>I_S = {h \nu \over \sigma(\nu) \cdot \tau_S }</math>
where
: <math>h</math> is the [[Planck constant]], and
: <math>\tau_\text{S}</math> is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification.
: <math>\nu</math> is the frequency in Hz
The minimum value of <math>I_\text{S}(\nu)</math> occurs on resonance,<ref>{{Cite book|last=Foot|first=C. J.|url=https://books.google.com/books?id=kXYpAQAAMAAJ|title=Atomic physics|date=2005|publisher=Oxford University Press|isbn=978-0-19-850695-9|pages=142|language=en}}</ref> where the cross section <math>\sigma(\nu)</math> is the largest. This minimum value is:
: <math>I_\text{sat} = \frac{\pi}{3}{h c \over \lambda^3 \tau_S }</math>
For a simple two-level atom with a natural linewidth <math>\Gamma</math>, the saturation time constant <math>\tau_\text{S}=\Gamma^{-1}</math>.

=== 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>

=== Small signal approximation ===
In the special case where the input signal is small compared to the saturation intensity, in other words,
: <math>I_{in} \ll I_S  \, </math>
then the general gain equation gives the small signal gain as
: <math> \ln(G) = \ln(G_0) = \gamma_0(\nu) \cdot z</math>
or
: <math> G = G_0 = e^{\gamma_0(\nu) z}</math>
which is identical to the small signal gain equation (see above).

=== Large signal asymptotic behaviour ===

For large input signals, where
: <math>I_{in} \gg I_S  \, </math>
the gain approaches unity
: <math>G \rightarrow 1 </math>
and the general gain equation approaches a linear [[asymptote]]:
: <math>I(z) = I_{in} + { \gamma_0(\nu) \cdot z \over \bar{g}(\nu) } I_S</math>