Finesse and quality factor

The finesse $F$ of the resonator is defined as the ratio of the free spectral range and the full width at half maximum of a resonance for a specific resonance wavelength. With the FSR and the FWHM given by Equations (1.11), (1.15), for the present model the finesse is given by

$$F = \frac{\Delta \lambda}{2\delta \lambda}= \pi \frac{\vert{\sf {... ...t{\sf {S}}_{\mbox{\scriptsize bb}}\vert^2 \mbox{e}^{\displaystyle - \alpha L}}.$$ (1.16)

The ability of the cavity to confine the field is described by the quality factor $Q$. It is a measure of the sharpness of the transmission peak and defined as the ratio of the resonance wavelength to the full width at half maximum ^1.2:

$$Q = \frac{\lambda}{2\delta \lambda}= \pi \frac{n_{\mbox{\scriptsi... ...a L}} = \frac{n_{\mbox{\scriptsize eff}}L_{\mbox{\scriptsize cav}}}{\lambda} F.$$ (1.17)

For a circular resonator with radius $R$ and cavity length $L_{\mbox{\scriptsize cav}}= 2 \pi R$, one obtains

$$Q = k R n_{\mbox{\scriptsize eff}} F$$ (1.18)

for the relationship between $Q$ and finesse $F$.

As before, the approximations according to Equations (1.9), (1.10) can be avoided by substituting the effective cavity mode index $n_{\mbox{\scriptsize eff}}$ in Equations (1.15), (1.17) by the effective group mode index $n_{\mbox{\scriptsize eff, g}}$ as defined in Equation (1.12).