The differential reflection phase shift, Δ = δp - δs, associated with the external reflection of monochromatic light at the surface of an absorbing medium is a monotonically decreasing function of the angle of incidence ø which is determined by the complex dielectric function ε. A new special angle of incidence, denoted by øΔ′max, is defined at which the slope Δ′ = ∂Δ/∂ø of the Δ-ø curve is maximum negative, Δ′max, and a transcendental equation is derived that determines this angle. øΔ′max differs from the principal angle øp at which Δ = 90°. As an example, øΔ′max is calculated by numerical iteration for light reflection at the air-Si interface for photon energies hv from 1.7 to 5.6eV in steps of 0.1eV, and is plotted, along with the associated maximum slope Δ′max, vs wavelength λ. It is noted that øΔ′max>øp at every λ, a result that may hold in general. Also, for 4.5 ≤ hv ≤ 5.6 eV, øΔ′max = 90°, so that a maximum negative slope occurs at grazing incidence in this spectral range. Another interesting observation is that, when |ε| >> 1 (e.g., for metals in the IR), Δ′(90°) is a direct measure of the extinction coefficient k = Imε½.
R. M. A. Azzam and A. M. El-Saba, "Maximum rate of change of the differential reflection phase shift with respect to the angle of incidence for light reflection at the surface of an absorbing medium," Appl. Opt. 28, 1365-1368 (1989)