Introduction
The Fabry-Perot instrument is a multiple-beam interferometer that is used in various forms for high-precision optical spectroscopy (cf. the Lummer-Gehrcke plate). In contrast to the two-beam Michelson interferometer one achieves sharp fringes on an extended dark background if the reflectivity of the mirror surface is chosen properly. The aim of this worksheet is to make the mathematics of this little wonder more transparent.
Almost any Optics text explains the subject with varying amounts of detail. One of them is Jurgen Meyer-Arendt: Introduction to Classical and Modern Optics , 2nd ed, Prentice-Hall 1984.
We used also: F.L. Pedrotti and L.S. Pedrotti: Introduction to Optics , Prentice-Hall 1987.
First of all, notice the difference between a plane-parallel plate (e.g., the Lummer plate), and a F-P cavity (two semi-transparent mirrors enclosing a cavity). Upon reflection the light beam undergoes a phase change when going from the optically thin to the optically dense medium. This phase change is important when considering constructive vs. destructive interference.
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Multiple beam interference from a parallel plate
Parallel plate:
Light is entering the plate at point A with angle of incidence a . The beam splits into a reflected wave (exit angle a ) with phase change (the plate is optically thicker than the surrounding medium), and a refracting wave (exit angle b ), which itself splits at the other edge of the plate (point C) into a reflected beam (no phase change, angle b ), and a refracted beam that exits with angle a .
One might think that the wave arriving at 1 is irrelevant, since there are many other beams, 2,3,4, etc. that are observed, when the parallel rays (that all originate from the single source ray 0) are brought together by a lens to interfere (the lens could be our eye, bringing the parallel rays to a point on the retina).
However, this is not so, and the situation depends on several parameters:
1) The subsequent internally reflected and then refracted beams (2,3,4) are attenuated depending on the transmission/reflection coefficients for the material (which in the case of the glass plate depend on the angle of incidence).
2) The plate thickness d , its length and the optics used to superimpose the parallel rays limit the number. In a Lummer plate the number of interfering beams is typically around 5-10.
In the Lummer-Gehrcke plate the phase shift effect is avoided by using a prism attached to the parallel plate to bring the light rays into the multiple reflection interference path with reflections near the critical angle (so that relatively little refraction takes place, and many passes of the beam interfere).
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