Extrema of the magnitude and the phase of a complex function of a real variable: application to attenuated internal reflection

Abstract

Given a complex function F(ω) = |F(ω)|exp[jΔ(ω)] of a real argument ω, the extrema of its magnitude |F(ω)| and its phase Δ(ω), as functions of ω, are determined simultaneously by finding the roots of one common equation, Im[G(ω)] = 0, where G = (F′/F)2 and F′ = ∂F/∂ω. The extrema of |F| and Δ are associated with Re G 0, respectively. This easy-to-prove theorem has a wide range of applications in physical optics. We consider attenuated internal reflection (AIR) as an example. In AIR the complex reflection coefficient for the p polarization, rp(ϕ), and the ratio of complex reflection coefficients for the p and s polarizations, ρ(ϕ) = rp(ϕ)/rs(ϕ), are considered as functions of the angle of incidence ϕ. It is found that the same (cubic) equation that determines the pseudo-Brewster angle of minimum |rp| also determines a new angle at which the reflection phase shift δp = arg rpexhibits a minimum of its own. Likewise, the same (quartic) equation that determines the second Brewster angle of minimum |ρ| also determines angles of incidence at which the differential reflection phase shift Δ = arg ρ experiences a minimum and a maximum. Angular positions and magnitudes of all extrema are calculated exactly for a specific case that represents light reflection by the vacuum–Al or glass–aqueous-dye-solution interface. As another example, the normal-incidence reflection of light by a birefringent film on an absorbing substrate is examined. The ratio of complex principal reflection coefficients is considered as a function of the film thickness normalized to the wavelength of light. The absolute value and the phase of this function exhibit multiple extrema, the first 13 of which are determined for a specific birefringent film on a Si substrate.