Analytic solution for the three-layer multiple beam interferometer

We present a simple analytic solution for the condition of constructive interference for light transmitted through an interferometer incorporating three ideally transparent layers of arbitrary thickness and refractive index. We also consider the effect of adding two metallic coatings to the outer surfaces of the interferometer and give empirical expressions for the associated phase changes for silver coatings on silica, sapphire, and mica substrates. A particular application to fringes of equal chromatic order can be utilized to obtain precise measurements of the thickness of extremely thin films sandwiched between two substrates.