A Fabry-Perot interferometer consists of two partially reflecting mirrors separated by a distance \(L\). Light entering the cavity is reflected back and forth many times, so the transmitted field is the coherent sum of many emerging beams. When the phase gained in one round trip matches a resonance condition, the transmitted waves reinforce each other strongly. When the phase is off resonance, the transmission drops.
This makes the Fabry-Perot much sharper than a simple two-beam interferometer. Instead of only two interfering waves, many internal reflections contribute, and the resulting transmission peaks can become very narrow if the mirrors are highly reflective.
For ideal mirrors with reflectivity \(r\), the standard finesse formula is
\[
F = \frac{\pi\sqrt{r}}{1-r}.
\]
Finesse is a dimensionless measure of how sharp the resonance peaks are. Higher reflectivity gives larger finesse, and larger finesse means narrower transmission peaks relative to the spacing between them.
Another important quantity is the free spectral range, the spacing between adjacent transmission resonances. In frequency form it is
\[
\Delta \nu = \frac{c}{2L},
\]
where \(c\) is the speed of light. This formula comes from the resonance condition for the cavity. At normal incidence, resonance occurs when an integer number of half-wavelengths fits inside the cavity:
\[
2L = m\lambda,
\qquad
m = 1,2,3,\dots
\]
That integer \(m\) is called the interference order. Near a given wavelength, the approximate spacing in wavelength between adjacent resonances is
\[
\Delta \lambda \approx \frac{\lambda^2}{2L}.
\]
The linewidth of each peak is roughly the free spectral range divided by the finesse:
\[
\delta \nu \approx \frac{\Delta \nu}{F},
\qquad
\delta \lambda \approx \frac{\Delta \lambda}{F}.
\]
This is why finesse matters so much: it controls the ratio of resonance spacing to resonance width. A high-finesse cavity has narrow, well-separated peaks.
A useful approximation for resolving power is
\[
R \approx mF,
\]
where
\[
m \approx \frac{2L}{\lambda}.
\]
So larger cavity length or higher finesse both improve resolution.
The transmission itself is described by the Airy formula. A common way to write it is
\[
T = \frac{1}{1+\mathcal{F}\sin^2(\delta/2)},
\]
where
\[
\mathcal{F} = \frac{4r}{(1-r)^2}
\]
is the Airy coefficient, and the round-trip phase at normal incidence is
\[
\delta = \frac{4\pi L}{\lambda}.
\]
Transmission peaks occur when \(\delta=2\pi m\), which is exactly the same as the resonance condition \(2L=m\lambda\).
For the sample input \(r=0.9\), \(L=1\ \text{cm}\), and \(\lambda=500\ \text{nm}\), the finesse is
\[
F = \frac{\pi\sqrt{0.9}}{1-0.9}
\approx 29.8.
\]
That matches the sample result. The order is
\[
m \approx \frac{2L}{\lambda}
= \frac{2(0.01)}{500\times10^{-9}}
= 40000.
\]
Since \(m\) is an integer here, the chosen wavelength is exactly resonant:
\[
2L = 40000\,\lambda.
\]
So the Airy transmission reaches a peak.
The animation in this calculator shows a schematic cavity with multiple internal reflections and several transmitted components emerging on the right. In reality, these transmitted beams combine coherently into a single transmitted field. The right-side plot shows how narrow resonance peaks appear around the nearest resonant wavelength.
At more advanced university level, one studies mirror losses, phase dispersion, intracavity media, angle dependence, and the distinction between ideal finesse and effective finesse. Those topics matter in laser resonators, spectroscopy, etalons, and optical filtering. But the central ideas remain:
\[
F = \frac{\pi\sqrt{r}}{1-r},
\qquad
\Delta\nu = \frac{c}{2L},
\qquad
T = \frac{1}{1+\mathcal{F}\sin^2(\delta/2)}.
\]
Together, these formulas explain why a Fabry-Perot cavity produces narrow transmission resonances and why high reflectivity leads to high spectral selectivity.
