Tool · Resonators

Fabry-Perot Cavity Calculator

How far apart are the resonances of a two-mirror cavity, how sharp is each one, and how long does light stay inside? The calculator takes the length, index, mirror reflectivities and internal loss of a plane-parallel or confocal cavity and reports the free spectral range, finesse, linewidth, Q, photon lifetime, peak transmission and fringe contrast, with the transmission and reflection spectra drawn from the exact field sums. Background: Fabry-Perot resonator, finesse, and free spectral range.

Cavity

Geometry

Length L

Index n, group index n_g

n sets the mode number, n_g the spacing

Mirror reflectivities R₁, R₂ (%)

Internal loss α_i

Wavelength λ₀ (nm)

Presets

Plane-wave theory for two lossless mirrors with an intracavity power loss α_i; mirror absorption can be folded into α_i. The confocal setting models a mode-degenerate confocal interferometer: light retraces its path after 4L and four reflections, so the free spectral range is c/4L, the round-trip factor is R₁R₂e^−2αL and the transmission is shared between two output spots. Curved-mirror stability, transverse modes and mode matching are outside the tool.

Readouts

Transmission and reflection versus detuning

transmissionreflection

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Finesse versus mirror reflectivity

lossless cavitywith the set internal loss

Checked against

These checks run in your browser on every load. The closed forms are compared with hand-computed values, the exact Airy linewidth with a numerical half-maximum search on the spectrum, the complex field sums with energy conservation, and the laser-cavity quantities with textbook numbers.

Check	Expected	Computed	Tolerance

The expected values are closed forms from Siegman, Lasers, chapter 11, and Hercher (1968) for the confocal interferometer, together with the laser-cavity relations of Coldren, Corzine and Mašanović, all evaluated by hand for the stated cavities. The tolerance is the largest relative difference from Expected that still passes.

The model

Light that enters through the first mirror makes repeated round trips, and each trip adds a partial wave to the transmitted field with the amplitude factor

\rho = \sqrt{R_1 R_2}\,e^{-\alpha_i L}

where $R_1$ and $R_2$ are the mirror power reflectivities and $\alpha_i$ the internal power loss. Summing the partial waves gives the Airy transmission

T(\nu) = \frac{T_{\max}}{1 + F\sin^2(\pi\nu/\mathrm{FSR})}

F = \frac{4\rho}{(1-\rho)^2}

with the free spectral range and the finesse

\mathrm{FSR} = \frac{c}{2 n_g L}

\mathcal{F} = \frac{\pi\sqrt{\rho}}{1-\rho}

The linewidth is found exactly from the half-maximum of the Airy function and is close to $\mathrm{FSR}/\mathcal{F}$ once the finesse exceeds about 10. The photon lifetime is the ring-down time of the stored energy,

\tau_p = \frac{n_g L}{c\,(\alpha_i L + \tfrac12 \ln\frac{1}{R_1R_2})}

and agrees with $1/2\pi\Delta\nu$ only when the resonance is close to a Lorentzian, which needs a finesse above about 10. In the confocal setting the tool models a mode-degenerate confocal interferometer: light retraces its path after $4L$ and four reflections, so the free spectral range is $c/4n_gL$ , the round-trip factor is $R_1R_2e^{-2\alpha_iL}$ , and the finesse is about half that of a plane cavity with the same mirrors while the linewidth and lifetime are the same. The theory is for plane waves and lossless mirrors (mirror absorption can be folded into $\alpha_i$ ); it does not include mirror curvature and stability, transverse modes, mode matching, surface figure or the finite aperture of real etalons, which lower the finesse a real device reaches.

Worked example

The default is a 1 cm air-spaced cavity with 99 % mirrors at 1550 nm. The free spectral range is $c/2L = 14.99\ \mathrm{GHz}$ , or 120.1 pm in wavelength. With $\rho = 0.99$ the finesse is $\pi\sqrt{0.99}/0.01 = 312.6$ , so each resonance is 47.95 MHz wide and the Q at 193.4 THz is $4.03\times10^{6}$ . Light survives for $\tau_p = 3.32\ \mathrm{ns}$ , about 50 round trips. The lossless, matched cavity transmits 100 % on resonance and 2.5 parts in $10^5$ between resonances, a contrast of 46.0 dB, and the forward wave inside is 100 times the input power. The nearest resonance to 1550 nm is at 1550.027 nm, mode number 12,903.

References: A. E. Siegman, Lasers, University Science Books (1986), chapter 11. M. Hercher, “The spherical mirror Fabry-Perot interferometer,” Applied Optics 7, 951–966 (1968). L. A. Coldren, S. W. Corzine and M. L. Mašanovi&cacute;, Diode Lasers and Photonic Integrated Circuits, 2nd ed., Wiley (2012), chapter 2.