Spontaneous Emission Coupling Factor

From Qwiki

Commonly denoted by beta ( $β$ ), it is the fraction of spontaneous emission coupled into the cavity mode of interest.

In terms of basic cavity QED parameters, cavity field loss rate $κ$ , spontaneous emission rate out the side of the cavity $γ$ , and atom-field coupling $g$ we have

$\beta={{g^2/\kappa}\over{\gamma/2+g^2/\kappa}}={{2C_1}\over{(1+2C_1)}}$

where one has the cooperativity parameter for a single atom

$C_1={{g^2}\over{\kappa\gamma}}={{1}\over{N_0}}$

where $N 0$ is the critical atom number The numerator is the rate of spontaneous emission into the cavity mode, showing the enhancement given by the Purcell factor, while the denominator is the the sum of spontaneous emission rates into the cavity mode and out the side of the cavity.
This result only holds in the bad cavity limit ( $2\kappa \gg \gamma(1+2C_1)$ ) where one can adiabatically eliminate the field variables, and irreversible spontaneous emission is well defined. For example in the strong coupling limit ( $g \gg \kappa,\gamma$ ) there are vacuum-Rabi oscillations. Also note that $γ$ is the spontaneous emission rate out the side of the cavity; it is generally less than the free space Einstein A coefficient by a factor of $ΔΩ / 4π$ where $ΔΩ$ is the solid angle subtended by the cavity mode.
Typical values for $β$ would be:

$10 - 5$ for a standard commerical diode laser
$0.1$ for a small semiconductor microdisk laser (diameter $D{\sim}{\lambda_{0}}$ )
$> 0.5$ for a photonic crystal microcavity with a wavelength-scale mode volume ( $V{\sim}(\lambda_{0}/n)^3$ )