Achieving a given reflectance for unpolarized light by controlling the incidence angle and the thickness of a transparent thin film on an absorbing substrate: application to energy equipartition in the four-detector photopolarimeter

At a given wavelength λ we determine all possible solution pairs (ϕ, ζ) of the incidence angle ϕ and the thickness ζ of a transparent thin film on an absorbing substrate that achieve a given unpolarized light reflectance. ${\mathcal{R}}_u$. The trajectory of the point that represents a solution pair in the ζ, ϕ plane depends on the optical properties of the film and substrate and on whether ${\mathcal{R}}_u$ is greater than or less than the normal-incidence reflectance ${\overline{\mathcal{R}}}_0$ of the bare substrate. When ${\mathcal{R}}_u>{\overline{\mathcal{R}}}_0$, the specified reflectance is achieved over a limited range of ϕ. At the least possible incidence angle, the film thickness is ≈⅛th wave. As an application we consider SiO₂ films on Si detectors that produce ${\mathcal{R}}_u=0.75$, 0.6667, and 0.50 at λ = 337 and 633 nm. If the first three detectors of the four-detector photopolarimeter (FDP) are coated to have these reflectance levels, with the reflectance diminishing in the direction of propagation of the light beam, and the last detector is antireflection coated (e.g., with a quarter-wave Si₃N₄ layer), equipartition of energy among the four detectors is accomplished for incident unpolarized light. Such a condition is desirable in the operation of the FDP. The ellipsometric parameters of the coated surfaces and the FDP instrument matrix are also calculated.