7 - Infinite series

Summary

Introduction: the Fabry–Perot interferometer

High precision interferometers typically employ reflecting surfaces to interfere an optical beam with itself multiple times. One of the earliest of these, developed by Charles Fabry and Alfred Perot, is also one of the most persistently useful and versatile interferometeric devices. First introduced in 1897 as a technique for measuring the optical thickness of a slab or air or glass [FP97], the device found its most successful application only two years later as a spectroscopic device [FP99]. In its simplest incarnation, the interferometer is a pair of parallel, partially reflecting and negligibly thin mirrors separated by a distance d; it is illustrated in Fig. 7.1. A plane wave incident from the left will be partially transmitted through the device, and partially reflected; the amount of light transmitted depends in a nontrivial way upon the properties of the interferometer, namely the mirror separation d, the mirror reflectivity r and transmissivity t, and the wavenumber k of the incident light. In this section we will study the transmission properties of the interferometer and show that a solution to the problem requires the summation of an infinite series.

The most natural way to analyze the effects of the Fabry–Perot is to follow the possible paths of the plane wave through the system and track all of its possible behaviors. We begin with a monochromatic plane wave incident from the left of the form U(z, t) = Aexp[ikz – iωt].