We can tell a lot about atomic structure and interactions by measuring spectra at very high resolution. One such example is shown below for a hydrogen/deuterium mixture. (Deuterium is a heavy isotope of hydrogen, with one proton and one neutron.) There is only a 0.2 nm average difference between the red lines of the two isotopes, and each isotope line is further split into a pair of "fine structure" lines with only 0.016 nm difference (caused by spin-orbit coupling). We want to design an etalon to measure these 4 lines to high precision. Therefore we will need an etalon with a free spectral range of 0.5 nm (to cover the range from 656.0 nm to 656.5 nm) and a linewidth of 0.001 nm (to measure the fine structure split).
(a) Find the thickness of the etalon you will need.
(b) Find the reflectivity R that you will need.
(c) Find the total number of distinct wavelengths that you can resolve with this etalon (this is the finesse ).
(d) Find the coefficient of finesse F, and plot the transmission function T vs. δ.
(e) Assuming the etalon is just a bare piece of glass, find the refractive index n needed for the reflectivity R that you found in part (b). (The value of n you will find is clearly unrealistic! In practice, high reflectivity is achieved by using metal coatings on both surfaces.)
Deuterium
Hydrogen
656.1 656.2 656.3
λ(nm)