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 phase Δ(ω) as functions of ω are determined simultaneously by finding the roots of one common equation, lm[G(ω)] = 0, where G = (F'/F)² and F' = ∂F/∂ω. The extrema of |F| and Δ are associated with ReG ᐸ 0 and ReG ᐳ 0, respectively. This easy-to-prove theorem has a wide range of applications in optics and electrical engineering. We offer at least one example of attenuated internal reflection (AIR). In AIR the complex reflection coefficient for the p polarization R_p(ϕ) and the ratio of complex reflection coefficients for the p and s polarizations, p(ϕ) =R_p(ϕ)/R_s(ϕ), are considered as functions of the angle of incidence ϕ. It is found that the same (cubic) equation¹ that determines the pseudo-Brewster angle of minimum |R_p| also determines a new angle at which the reflection phase shift δ_p = argR_p exhibits a minimum of its own. Likewise, the same (quartic) equation² that determines the second Brewster angle of minimum |_ρ| also determines angles of incidence at which the differential reflection phase shift Δ = argρ experiences a minimum and a maximum. Angular positions and magnitudes of all extrema are exactly calculated for a specific case that represents light reflection by the vacuum —Al or glass—aqueous dye solution interface.

© 1987 Optical Society of America