Fabry-Perot Etalon (2D)

What you are looking at

A Fabry-Perot etalon is two parallel, partly reflecting mirrors a small distance L apart. Light bounces back and forth between them, and a fraction leaks out of the second mirror on every pass. These transmitted beams are added together. When the extra path of one round trip, 2 n L cos(theta), is a whole number of wavelengths, all the beams are in step and nearly all the light gets through. Otherwise they cancel. The result is a set of very sharp transmission peaks: the sharper the mirrors, the sharper the peaks.

Equations used

• Round-trip phase delta = 4 pi n L cos(theta') / lambda, where sin(theta) = n sin(theta') links the angle outside (theta) and inside (theta') the gap.
• Airy transmission T = T_m^2 / ((1 - R)^2 + 4 R sin^2(delta/2)) with mirror transmission T_m = 1 - R - A (A = absorption). Without loss this is 1 / (1 + F sin^2(delta/2)) with the coefficient of finesse F = 4R/(1-R)^2.
• Free spectral range FSR = c / (2 n L) in frequency, or lambda^2 / (2 n L) in wavelength: the spacing between neighbouring peaks.
• Finesse pi sqrt(F) / 2 = FSR / FWHM. With flat-surface errors of lambda/N the finesse is limited to about N/2, combined as 1/F_eff^2 = 1/F_R^2 + 1/(N/2)^2; the page smooths the Airy function with a Gaussian of that width.
• Resolving power lambda / d(lambda) = m F_eff with order m = 2 n L / lambda. Two lines are separated when their spacing exceeds the line width (Taylor criterion; Rayleigh-like limit near 0.8 of it).
• Ring pattern: for an extended source and a lens, the screen radius is f * theta, and bright rings occur where delta is a multiple of 2 pi, with radii growing like the square root of the ring number.

What every control does

• Mirror reflectivity sets the finesse: higher R gives thinner rings and a longer effective beam path. Absorption lowers the peak transmission. Flatness shows how rough mirrors blur the rings.
• Spacing and refractive index set the order, the FSR and the ring spacing. A longer gap gives finer peaks and a finer resolution but a smaller FSR. Tilt moves the ring centre (and changes the angle for the photodiode).
• Wavelength and second line add a close doublet (the sodium D lines are 0.597 nm apart); the ring screen and the spectrum graph show whether it is resolved. Field of view zooms the ring screen.
• Piezo scan moves one mirror by a few hundred nanometres, which is only about one wavelength: the rings breathe, and in the collimated mode the photodiode signal traces the Airy peaks. Observation switches between the extended-source ring screen and a laser on a photodiode.

Honest simplifications

Plane parallel mirrors, a single wavelength per line and no phase shift on reflection (it only offsets the peaks). The flatness is modelled as a Gaussian broadening, not a surface map; walk-off, mirror curvature, finite aperture and beam divergence are ignored. The ring brightness uses the ring colour of the wavelength, not a calibrated spectrum. The 3D page exaggerates the mirror gap so you can see it move.

Checks run in the page

(1) A direct sum of the transmitted beam amplitudes (thousands of round trips) matches the Airy formula at nine phases. (2) The measured full width at half maximum gives the same finesse as pi sqrt(F)/2. (3) For a lossless etalon the reflected and transmitted powers, both from summed beams, add up to one.