The diagram shows two rays emerging from the Fabry-Perot, one of which, To, passes through the etalon without reflections, and the other, T[, is reflected twice by the coatings, before emerging. The refractive index of the medium between the plates is n, and that of the medium outside the gap is no. In our case the medium outside the gap is the material of the plates themselves, quartz, and the emerging rays have to pass through another refraction before they finally leave the etalon, but the pattern of the resulting transmission is unaffected by this, so it is ignored in this analysis.
At each reflection, the amplitude is reduced by
and the phase is shifted by
while at each transmission through an interface the amplitude is reduced by
Assuming no absorption, we have, by the law of conservation of energy, T + R = 1.
Using complex form for a light ray, and defining the amplitude of the wave at point (a) as unity, the amplitude at point (b) is
where k is the wave number inside the etalon, given by
is the vacuum wavelength.
At point (c) the amplitude is
The total amplitude of the two beams will be the sum of their amplitudes, measured at a wave front defined by a line perpendicular to the direction of the beam. We therefore need to add, to the amplitude at point (b), a modified amplitude, equal in magnitude to Ti, but retarded in phase by an amount
where
is the wave number outside of the etalon, given by
Thus
where lo is given by
Neglecting the
phase change due to the two reflections, we have for the phase difference between the two beams
From the law of refraction
So the phase difference may be written
To within a constant beam can be written as multiplicative phase factor, the amplitude of the m emergent
The total transmitted beam is the sum of all individual beams
This is the sum of a geometric series, and so can be written as
The intensity of the beam is now
and, since the incident beam was assumed to have an intensity of unity, this will also give the transmission function:
This completes the derivation of equation 2.1.
