The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich's method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich's method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich's method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names.The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.

We present an ab initio account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the description of Gaussian beam diffraction as compared with full-wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lenslike medium, the CGO equations possess analytical solutions; otherwise, they can be readily solved numerically. As a nontrivial example we consider Gaussian beam propagation and diffraction along a helical ray in an axially symmetric waveguide medium. It is shown that the major axis of the beam's elliptical cross section grows unboundedly; it is oriented predominantly in the azimuthal (binormal) direction and does not obey the parallel-transport law.

Isoflavones genistein and daidzein are nonsteroidal phytoestrogens occurring mainly in soybean foods. These phytoestrogens possess estrogenic properties and show a variety of health benefits as anti-inflammatory agents. However, the mechanism of their action has not been identified in detail. The aim of this study is to characterize the antioxidant powers of genistein, daidzein and daidzein metabolite-equol through their activities to scavenge superoxide anion radical (O(•) 2 (-) ), hydroxyl radical (HO(•) ), 2,2-diphenyl-1-picrylhydrazyl radical (DPPH(•) ) and hydrogen peroxide (H2 O2 ) using chemiluminescence and spectrophotometry techniques. Potassium superoxide in dimethyl sulphoxide (DMSO) and 18-crown-6 ether were used as a source of O(•) 2 (-) . Hydroxyl radicals were produced using the Fenton reaction. In free radical assays, genistein had the IC50 values (an amount of antioxidant concentration required to decrease the initial radical concentration by 50%) 0.391 ± 0.012 mM for O(•) 2 (-) , 0.621 ± 0.028 mM for HO(•) and 1.89 ± 0.16 mM for DPPH(•) . The IC50 values for daidzein for these free radicals were 1.924 ± 0.011 mM, 0.702 ± 0.012 mM and 2.81 ± 0.03 mM, respectively. Equol was the most active the free radical scavenger with IC50  = 0.451 ± 0.018 mM for HO(•) and IC50  = 1.36 ± 0.11 mM for DPPH(•) . All tested compounds exerted a significant effect on the H2 O2 : IC50  = 18.1 ± 1.1 μM for genistein, IC50  = 2.1 ± 0.5 μM for daidzein, and IC50  = 1.06 ± 0.2 μM for equol. These findings show that genistein, daidzein and equol are effective free radical scavengers and possess high antioxidant power in vitro. Copyright © 2016 John Wiley & Sons, Ltd.

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