Fabry–Perot resonances and finesse

Let each identical lossless mirror have intensity reflectivity $R=|r|^2$ and intensity transmissivity $1-R$, with equal external impedances and no absorption. The magnitude of the single-mirror field reflection coefficient is $\sqrt R$. A complete internal round trip encounters two mirrors, so its field attenuation magnitude is $|r_1r_2|=R$, while its surviving intensity fraction is $R^2$. These three quantities refer to different operations and should never share an unqualified symbol.

Define $\delta$ as the total round-trip phase including any constant reflection phases. The first transmitted field is followed by copies multiplied by $Re^{i\delta}$, so the coherent field series is geometric. Its denominator becomes small near phase agreement. A symmetric lossless cavity has unit peak transmission even when each mirror transmits little individually: the steady internal field builds until the emerging fields satisfy both interference and energy conservation. Loss, unequal mirrors, or mode mismatch can reduce that peak.

For normal incidence in a nondispersive cavity of length $L$ and refractive index $n$, round-trip propagation phase grows as $4\pi nL\nu/c$. Increasing frequency until this phase advances by $2\pi$ reaches the next longitudinal resonance. The resulting free spectral range is a frequency spacing, not a linewidth. Strong material or mirror dispersion changes the derivative of round-trip phase, so a group-delay calculation must replace the simple nondispersive expression.

Near a high-reflectivity peak, approximate the sine of the small phase detuning by its argument. The half-maximum points give a full phase width approximately $2(1-R)/\sqrt R$. Dividing the $2\pi$ resonance spacing by this full width gives finesse $\pi\sqrt R/(1-R)$. The square root appears because R is an intensity reflectivity. This is a narrow-peak approximation; using a half width as the full width introduces a separate factor-of-two error.

High reflectivity lets many delayed copies contribute appreciably, extending the time over which the cavity remembers a coherent input. A longer temporal response corresponds to a narrower spectral response. This is a relation between the steady frequency response and the passive decay of a mode, not evidence that changing mirror reflectivity during a run implements the same stationary experiment. A rapidly scanned source may also fail to reach the steady Airy response if the scan outruns the cavity settling.

Increasing cavity length at fixed mirrors reduces both free spectral range and linewidth by the same factor, leaving ideal finesse unchanged. Increasing reflectivity instead raises finesse and narrows the peaks relative to their spacing. Finesse is therefore different from resolving power at an optical carrier, which compares that carrier frequency with a linewidth. An instrument can have closely packed resonances, and order ambiguity must still be addressed when using it to interpret an unknown spectrum.