Abstract

For any optically isotropic bare or coated substrate the power reflectance R_v(ϕ) is an even function, R_v(–ϕ) = R_v(ϕ), of the angle of incidence ϕ, hence all the odd-numbered derivatives $R_v^{\left(n\right)}=d^n{R}_v/d_{\varphi }^n$ are identically 0 at ϕ = 0, independent of the incident polarization v. For a bare substrate and incident unpolarized light (v = u), the second derivative $R_u^{\left(2\right)}$ also, so that the flatness of the R_u vs ϕ curve, from ϕ = 0 up to a large angle (>45°) is determined by the fourth derivative $R_u^{\left(4\right)}$. The condition that $R_u^{\left(4\right)}=0$ gives the maximally flat response (MFR) and leads to a specific constraint between the real and imaginary parts, ϵ_r and ϵ_i, of the substrate complex dielectric function ϵ. This constraint is determined and the corresponding complex plane locus is presented. Examples of specific substrates with the MFR at specific wavelengths are given. Subsequently, the case of an arbitrary absorbing substrate which is coated by a transparent thin film is considered. The refractive index and thickness of the film that null $R_u^{\left(2\right)}$ and $R_u^{\left(4\right)}$ simultaneously are determined to achieve the MFR by the film-substrate system at a given wavelength.

© 1989 Optical Society of America