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Authors: Settimi, A.* 
Sciacca, U.* 
Bianchi, C.* 
Title: Scientific review on the Complex Eikonal, and research perspectives for the Ionospheric Ray-tracing and Absorption
Other Titles: Rassegna scientifica sull’Iconale Complessa e prospettive di ricerca per il Ray-tracing e l’Assorbimento ionosferici
Issue Date: 20-Mar-2013
Series/Report no.: Quaderni di Geofisica
Keywords: Ionosphere
Quasi Longitudinal propagation
Non-Deviative absorption
Complex Eikonal theory
Subject Classification01. Atmosphere::01.02. Ionosphere::01.02.04. Plasma Physics 
01. Atmosphere::01.02. Ionosphere::01.02.05. Wave propagation 
05. General::05.05. Mathematical geophysics::05.05.99. General or miscellaneous 
Abstract: The present paper conducts a scientific review on the complex eikonal, extrapolating the research perspectives on the ionospheric ray-tracing and absorption. As regards the scientific review, the eikonal equation is expressed, and some complex-valued solutions are defined corresponding to complex rays and caustics. Moreover, the geometrical optics is compared to the beam tracing method, introducing the limit of the quasi-isotropic and paraxial complex optics approximations. Finally, the quasi-optical beam tracing is defined as the complex eikonal method applied to ray-tracing, discussing the beam propagation in a cold magnetized plasma. As regards the research perspectives, this paper proposes to address the following scientific problem: in absence of electromagnetic (e.m.) sources, consider a material medium which is time invariant, linear, optically isotropic, generally dispersive in frequency and inhomogeneous in space, with the additional condition that the refractive index is assumed varying even strongly in space. The paper continues the topics discussed by Bianchi et al. [2009], proposing a novelty with respect to the other referenced bibliography: indeed, the Joule’s effect is assumed non negligible, so the medium is dissipative, and its electrical conductivity is not identically zero. In mathematical terms, the refractive index belongs to the field of complex numbers. The dissipation plays a significant role, and even the eikonal function belongs to the complex numbers field. Under these conditions, for the first time to the best of our knowledge, suitable generalized complex eikonal and transport equations are derived, never discussed in literature. Moreover, in order to solve the ionospheric ray-tracing and absorption problems, we hint a perspective viewpoint. The complex eikonal equations are derived assuming the medium as optically isotropic. However, in agreement with the quasi isotropic approximation of geometrical optics, these equations can be referred to the Appleton-Hartree’s refractive index for an ionospheric magneto-plasma, which becomes only weakly anisotropic in the presence of Earth’s magnetic induction field. Finally, a simple formula is deduced for a simplified problem. Consider a flat layering ionospheric medium, so without any horizontal gradient. The paper proposes a new formula, useful to calculate the amplitude absorption due to the ionospheric D-layer, which can be approximately modelled by a linearized complex refractive index, because covering a short range of heights, between h1= 50 km and h2= 80 km about.
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