General expression for the reflectance of an all-dielectric multilayer stack

A converted characteristic matrix method has been used to derive a generalized expression for reflectance R (for wavelength lambda at normal incidence) of an all-dielectric multilayer stack consisting of K layers of refractive indices n(1),n(2),n(3)... n(k-1),n(k) having quarterwave thicknesses at the monitoring wavelength lambda(0). The expression thus derived is of the form R = Sigma(K)(N=0)(P(N) tan(2N)?)/(Q(N)tan(2N)?), 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 ? = {(Pi/2) [(lambda - lambda(0))/lambda]} which assumes positive or negative values depending on whether lambda < lambda(0) or A > lambda(0) such that within limits of acceptable variations (Deltalambda) in lambda(0), R remains unaffected at +/-?. 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.