Fabry-Perot Etalon Spectrometer

Two parallel mirrors of reflectance $R$ a distance $d$ apart pass light only through multiple-beam interference, giving the Airy transmission $T=1/(1+F\sin^2(\delta/2))$ with $F=4R/(1-R)^2$ and round-trip phase $\delta=4\pi d/\lambda$ . As an instrument the etalon resolves two close wavelengths when its reflectance finesse $\mathcal{F}=\pi\sqrt R/(1-R)$ makes the Airy peaks narrower than the line separation, with resolving power $R_p=m\mathcal{F}$ , provided the free spectral range $\lambda^2/2d$ exceeds the separation so the orders do not overlap. Here the test source is the sodium doublet (D2 588.995 nm, D1 589.592 nm): raise $R$ and watch the single blurred peak split cleanly into two.

Figure 1. The Fabry-Perot etalon: the multiple-reflection schematic and the ring fringe pattern (top), and the Airy combs of the two sodium D lines and their sum as the spacing is scanned (bottom). All sharpen as the reflectance rises. Method: closed-form Airy multiple-beam transmission; the schematic shows the geometric beam cascade and the rings are T versus angle.

reflectance R0.60

spacing d (um)80

line gap dl (nm)0.60

WHAT TO TRY

Raise the mirror reflectance R: the Airy transmission peaks sharpen into needle-thin lines, the finesse climbing, so the etalon resolves ever finer wavelength differences.
Bring two spectral lines together: they merge once their peaks overlap, and the resolving power, finesse times order, is the limit on telling them apart.
Mind the free spectral range: the comb of transmission peaks repeats, so a Fabry-Perot is a ruler that is only unambiguous within a single order.