Relation between coupled-mode theory and equivalent layers for multilayer interference coatings

The method of equivalent layers is a commonly used technique for designing optical multilayer interference coatings. Herpin's theorem [C. R. Acad. Sci. 225, 182 (1947)] states that every symmetrical multilayer structure is equivalent, at one arbitrary wavelength, to a single homogeneous layer. The Herpin equivalent layer is described by two design parameters, the equivalent index and the equivalent thickness. Alternatively, we recently developed an exact coupled-mode analysis for the description of multilayer interference coatings composed of a symmetrical combination of layers. The design parameters of the coupled-mode theory are the exact coupling coefficient and the exact detuning coefficient. Recently we used this method in the design of chirped mirrors for dispersion compensation. We prove that the two methods are equivalent and derive relations that link the design parameters of both formalisms. By use of these relations it is possible to translate between the coupled-mode formalism and the method of equivalent layers. The simultaneous availability of both design methods gives a new perspective on the analytical design of optical interference coatings with challenging spectral response characteristics.