Abstract

A converted characteristic matrix method has been used to derive a generalized expression for reflectance R (for wavelength λ at normal incidence) of an all-dielectric multilayer stack consisting of K layers of refractive indices n₁,n₂,n₃ …n_k−1,n_k having quarterwave thicknesses at the monitoring wavelength λ₀. The expression thus derived is of the form $R={\sum }_{N=0}^K(P_N\hspace{0.17em}{\text{tan}}^{2N}\varphi )/(Q_N\hspace{0.17em}{\text{tan}}^{2N}\varphi )$, where P_N and Q_N are functions of refractive indices and their interdependence is expressed by the relation (Q_N − P_N)/(4n_s) = (K!)/[N!(K − N)!]. This expression shows that R is a function of (i) the refractive indices of the various layers in the multilayer structure and (ii) the parameter ϕ = {(π/2)[(λ − λ₀)/λ]} which assumes positive or negative values depending on whether λ > λ₀ or λ < λ₀ such that within limits of acceptable variations (Δ λ) in λ₀, R remains unaffected at ±ϕ. An outcome of this analysis is that it can be used in identifying, to a limited extent, the structure of an unknown optical coating once the experimentally observed parameters R and T are known.

© 1988 Optical Society of America

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