Because the emission rate at a given wavelength is directly proportional to the optical mode density (see Eq. 14.7), the emission rate enhancement spectrum is given by the ratio of the 1D cavity mode density to the 1D free space mode density. As calculated earlier, the cavity enhancement spectrum has a lorentzian lineshape. The enhancement factor at the resonance wavelength is thus given by the ratio of the optical mode densities with and without a cavity, i. e.

The equation shows that a strong enhancement of the spontaneous emission rate along the cavity axis can be achieved with microcavities.

Equation (14.19) represents the average emission rate enhancement out of both reflectors of the cavity. To find the enhancement in a single direction, we multiply the enhancement given by Eq. (14.19) by the fraction of the light exiting the mirror with reflectivity R1 (i. e. 1 - R1) divided by the average loss of the two mirrors for one round trip in the cavity (i. e. (1/2) [(1 - R1) + (1 - R2)] ). For large R1 and R2, this gives for the enhancement of the emission exiting R1

^ 2(1 - Rl) 2F ^ 1 - Rl 2F ^ 2 п(RlR2 )l/4 (1 - Rl)

2 - Ri - R2 п 1 - л IR1R2 п

where we used the approximation 1 - ( R1R2 )1/2 « (1/2) (1 - R1R2) « (1/2) (2 - R1 - R2). Equation (14.20) represents the emission rate enhancement from a single reflector with reflectivity R1.

Next we take into account the standing wave effect, that is, the distribution of the optically active material relative to the nodes and antinodes of the optical wave. The antinode enhancement factor £ has a value of 2, if the active region is located exactly at an antinode of the standing wave inside the cavity. The value of £ is unity if the active region is smeared out over

many periods of the standing optical wave. Finally, £ = 0 if the active material is located at a node.

The emission rate enhancement is then given by

Ge = f - 74--^)‘_Vj^ (14.21)

2 n (1 - (RRT )2 т

where R‘ is the reflectivity of the light-exit mirror and therefore R‘ < R2. Equation (14.21) also takes into account changes in the spontaneous emission lifetime in terms of т, the lifetime without cavity, and Tcav, the lifetime with cavity. The factor of Tcav / т ensures that the enhancement decreases if the cavity lifetime is reduced as a result of the cavity. For planar microcavities, the ratio of the spontaneous lifetime with a cavity, Tcav, and the lifetime without a cavity, т, is Tcav / т > 0.9 (Vredenberg et al., 1993). Thus, the emission lifetime is changed by only a minor amount in a planar microcavity.