Limaçon of Pascal locus of the complex refractive indices of interfaces with maximally flat reflectance-versus-angle curves for incident unpolarized light

Abstract

For an interface between two isotropic media the power reflectance Rν(ϕ) is an even function, Rν(ϕ) = Rν(−ϕ) of the angle of incidence ϕ; hence all the odd derivatives, Rν(n) = dnRν/dϕn (n odd), are identically 0 at ϕ = 0, independent of the incident polarization ν. When the incident light is unpolarized (ν = u), the second derivative, Ru(2) is also 0 at ϕ = 0, so that the flatness of the Ru-versus-ϕ curve over an initial range of ϕ starting from ϕ = 0 is determined by the fourth derivative, Ru(4). The condition that Ru(4) = 0 at ϕ = 0 gives the maximally flat response and leads to a specific constraint on the complex relative refractive index N, namely, that Re[(N − 1)2/N3] = 2/NN*. The corresponding complex plane contour is the limaçon of Pascal, η=2 cos θ±3 in polar form, where N = η exp(jθ). The two branches of this contour constitute the boundary lines that separate the region of the complex plane in which the function Ru(ϕ) is monotonic from that in which the function exhibits a minimum at oblique incidence. Families of curves that illustrate the maximally flat response in external and internal reflection are presented. New equations that determine the angle of incidence of minimum unpolarized-light reflectance of a dielectric–dielectric or a dielectric–conductor interface are derived.