KEYWORDS: Maxwell's equations, Electromagnetism, Electrodynamics, Probability theory, Information science, Information technology, Physics, Optical properties, Basic research
The most common physical formalisms are the Lagrangian formalism and the Hamiltonian formalism. From the superficial point of view, they are one and the same, but rewritten in other terms. However, it seems that the Hamiltonian formalism has a richer structure and is more convenient for studying the electromagnetic field, especially in the formalization of its geometrization. Unfortunately for field problems, there is a whole set of Hamiltonian formalisms. The authors study the applicability of different variants of the Hamiltonian formalism to the problems of electrodynamics. In this paper we consider the Hamiltonian formalism with constraints.
The Maxwell equations have a fairly simple form. However, finding solutions of Maxwell’s equations is an extremely difficult task. Therefore, various simplifying approaches are often used in optics. One such simplifying approach is to use the approximation of geometric optics. The approximation of geometric optics is constructed with the assumption that the wavelengths are small (short-wavelength approximation). The basis of geometric optics is the eikonal equation. The eikonal equation can be obtained from the wave equation (Helmholtz equation). Thus, the eikonal equation relates the wave and geometric optics. In fact, the eikonal equation is a quasi-classical approximation (the Wentzel–Kramers–Brillouin method) of wave optics. This paper shows the application of geometric methods of electrodynamics to the calculation of optical devices, such as Maxwell and Luneburg lenses. The eikonal equation, which was transformed to the ODE system by the method of characteristics, is considered. The resulting system is written for the case of Maxwell and Luneburg lenses.
Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrized optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrized approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects, Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated.
KEYWORDS: Maxwell's equations, Optical components, Electromagnetism, Physics, Radio propagation, Electrodynamics, Magnetism, Information science, Information technology, General relativity
The paper considers the technics of construction of optical devices based on the method of geometrization of Maxwell's equations. The method is based on representation of material equations in the form of an effective space-time geometry. Thus we get a problem similar to that of some bimetric theory of gravity. That allows to use a well-developed apparatus of differential geometry. On this basis, we can examine the propagation of the electromagnetic field on the given parameters of the medium. It is also possible to find the parameters of the medium by a given law of propagation of electromagnetic fields.
We investigate the waveguide propagation of polarized monochromatic light in a smoothly irregular transition between two regular planar dielectric waveguides. The single-mode approximation of the cross-sections method is used. The smooth evolution of the electromagnetic field propagating mode is calculated. The calculation is performed using the regularized stable numerical method.
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