Electromagnetic Modes of Maxwell Fisheye Lens

We provide an analysis of the radial structure of TE and TM modes of the Maxwell fisheye lens, by means of Maxwell equations as applied to the fisheye case. Choosing a lens of size R = 1 cm, we plot some of the modes in the infrared range. Pacs Numbers: 03.50.De, 42.79.Ry internet: [email protected] Electromagnetic Modes of Maxwell Fisheye Lens Haret Rosu Instituto de F́ısica, Universidad de Guanajuato, Apdo Postal E-143, 37150 León, Gto, México Marco Reyes Departamento de F́ısica, CINVESTAV, Apdo Postal 14-740, 07000 México D.F., México Maxwell fisheye lenses [1] are spherical inhomogenous spheres which belong to the class of complete, perfectly focussing optical systems, that is, within geometrical optics, (i) all rays are circles lying in planes containing the centre of the lens (ii) every point P has a conjugate P ′ (iii) the imaging is an inversion, OP × OP ′ = R2 (there are no spherical aberrations). Due to such good geometrical qualities, it is no wonder to be used in animal vision. At the present time people are trying to reproduce fisheye lenses by the so-called GRIN (graded index) technology [2]. The fisheye first appeared in Physics in the past century as a problem of the Irish Academy looking for the refractive index of a medium that could conceivably form images in the least depth. It was solved by Maxwell in 1854. The fisheye lens is one of the most symmetric systems one could find in Nature, and has been called ”the hydrogen atom of optics” [3]. Most studies considered only the geometrical optics approximation (eikonal theory). The similarities to the Kepler/Coulomb problem have been pointed out a long time ago [4]. These lenses have also interesting connections with phenomenological electrodynamics in curved space. Moreover, one can draw an analogy between the optical properties of standard Maxwell lenses and those of the synchronous gauge k = 1 of de Sitter spacetime (also called de Sitter-Lanczos model) in its expanding stage [5]. Besides, there are many interesting open problems concerning the propagation of signals in inhomogeneous media that were recently touched upon in the literature [6]. Due to their aberration-free imaging over a wide range of angles, inhomogeneous spherical lenses are important for antennas such as required in radar technology. In 1958, Tai [7] discussed the Maxwell equations in the fisheye case. At the end of his paper, Tai complained about the missing of computers allowing detailed numerical studies of his theoretical treatment. After many years, our work is just a step in this direction, complemented with some more comments. Tai parametrized the spherical symmetric index of