# Fabry-Perot Finesse vs Ring Resonator

Why is the finesse of a fabry-perot resonator given by $$2\pi/\text{RTL}$$ while the finesse of a ring resonator is given by $$\pi/\text{RTL}$$?

The finesse of a fabry-perot resonator is given by $$F = 2\pi/-\ln(R_1R_2)$$. (source: wikipedia) For high enough reflectivity $$-\ln(R_1R_2) \approx 1 - R_2 R_2$$, so $$F \approx 2\pi/\text{RTL}$$.

The finesse of a waveguide ring resonator is given by $$F = \pi/\kappa^2 \approx \pi/\text{RTL}$$. (source: Integrated Ring Resonators by D. G. Rabus, Chapter 2)

In Siegman's Lasers, Eq 55 of Chapter 11, he derives Finesse for a resonant cavity very similarly to Rabus' derivation, but with $$g_{rt}$$ instead of $$t^2$$. Siegman writes $$F=\frac{\pi \sqrt{g_{rt}}}{1-g_{rt}} \approx \frac{2\pi}{\delta_c - \delta_m}$$ whereas Rabus writes $$F=\pi\frac{t}{1-t^2} \approx \frac{\pi}{\kappa^2}.$$ (Siegman's $$\delta_c$$ is for cavity loss, but $$\delta_m$$ is for material loss/gain, not mirror loss.) So where does the factor of 2 come from? Is there an elegant way to account for it?

• I wonder if it’s related to the factor of 2 difference in the free spectral range Aug 7, 2020 at 14:10
• But don't they have the same FSR formula? $FSR = c/\text{OPL}$, right? How is it relevant that OPL=2nL (FP) vs OPL=nL (RR)? Oct 9, 2020 at 0:42

Rely on $$F = \frac{2\pi}{\text{RTL}}$$, where RTL is the TOTAL round trip power loss.

In the ring resonator,

$$\text{RTL}\approx\kappa^2 + \text{material loss} = 2\kappa^2$$

So $$F=\frac{2\pi}{2\kappa^2}=\frac{\pi}{\kappa^2}$$.

The reading of Siegman is wrong. Siegman's $$g_{rt}$$ is round trip field gain, and it is not analogous to $$t^2$$, but $$\delta$$ is power loss rate per round trip. So $$g_{rt}$$ is the square root of the round trip power factor: $$g_{rt} = \exp(-\delta/2)$$.

$$1-g_{rt}=1-exp(-\delta/2)\approx (1-(1-\delta/2)) = \delta/2$$

So Siegman recovers the factor of 2 missing from the numerator: $$F=\frac{\pi g_{rt}}{1-g_{rt}}=\frac{2\pi}{\delta}$$.

(Both sources ignore the factor of $$t$$ or $$\sqrt{g_{rt}}$$ in the numerator.)