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Approximate formulas
Published in Subodh Kumar Sharma, Elastic Scattering of Electromagnetic Radiation, 2018
An obstacle is termed soft if it satisfies the condition |m - 1| ≪ 1. Two approximation methods have been extensively used for predicting the scattering of electromagnetic waves by this class of scatterers. These are (i) the Rayleigh‐Gans approximation (RGA) and (ii) the anomalous diffraction approximation (ADA). The latter approximation method is closely related to another approximation, known as the eikonal approximation (EA). A detailed exposition to scattering of light by soft particles, including approximations, can be found in a monograph by Sharma and Somerford (2006).
Interaction of Radiation with Plasmas
Published in R A Cairns, A D R Phelps, P Osborne, Generation and Application of High Power Microwaves, 2020
In practice we invariably deal with inhomogeneous plasmas, so it is essential to consider wave propagation in this case. The equilibrium plasma can generally be regarded as time-independent, since any time variation of the equilibrium may be taken to be over a time scale much longer than the wave period. The wave can then be taken to propagate with constant frequency, imposed by the wave generator. The linearised Vlasov equation, which must be used if resonant damping is to be described, is difficult to solve in a general inhomogeneous system, and where it can be solved may lead to very complicated equations for the field. Various simpler techniques have been developed, of which one which is very useful is the ray-tracing or eikonal approximation. In essence this assumes that the wave propagates as a packet in which the wavenumber changes slowly, obeying the local dispersion relation. A single dispersion relation does not, of course, determine the three components of the wavevector, all of which may vary in a plasma with three dimensional geometry. However, it can be shown that the position of the wave and its wavenumber evolve according to the equations: drdt=−∂D∂k/∂D∂ω,dkdt=∂D∂r/∂D∂ω,where D(r,k, w) = 0 is the local dispersion relation. The initial conditions are determined by the position of the antenna which launches the wave and the spectrum of wavenumbers which it generates. Generally a bundle of rays, distributed across the antenna and covering the k spectrum, has to be looked at. Absorption is found by calculating the imaginary part of the component of k along the direction of the group velocity, which determines the damping of the wave along its path.
Geometric theory of topological defects: methodological developments and new trends
Published in Liquid Crystals Reviews, 2021
Sébastien Fumeron, Bertrand Berche, Fernando Moraes
The geometric description of linear defects revealed wedge disclinations carry curvature along their axis: a positive Frank angle corresponds to a conical geometry that focuses incoming light rays, in a similar fashion to what happens with a converging lens. Conversely, a negative Frank angle corresponds to a saddle-like geometry that scatters geodesics, in the same fashion as a diverging lens. It is worth noticing that the effect of a disclination does not limit to the eikonal approximation, as it also diffracts incoming waves: from the present geometric approach, computing the differential scattering cross-section of a wedge disclination showed good agreements with theoretical results obtained by Grandjean from standard acoustics [165,166].