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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?

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  • $\begingroup$ I wonder if it’s related to the factor of 2 difference in the free spectral range $\endgroup$
    – Jagerber48
    Aug 7, 2020 at 14:10
  • $\begingroup$ 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)? $\endgroup$ Oct 9, 2020 at 0:42

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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.)

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