Resonant Cherenkov type radiation in dispersive metamaterials

The radiation of moving charged particles makes a fundamental problem in modern physics and has long been the subject of a thorough study [1-26]. As early as 1889, O. Heaviside considered the electromagnetic field of a point charge uniformly moving in a dispersion-less medium and was seemingly the first (as follows from the literature) to ask himself: What happens when the charged particle velocity, u exceeds the velocity of light v in the medium? He found that the spherical wave front of particle radiation takes a conical shape with the semi-angle θ = ν/u reported in his Theory of Electromagnetism, volume 3, 1905 [1]. Later, Lord Kelvin got also curious about the matter and proposed in his lecture on thermal radiation [2], in 1900 that an atom moving at a speed exceeding the speed of light produces a conical wave, as it holds upon Mach principle in acoustics. In 1904, A. Sommerfeld considered a purely theoretic and even hypothetic problem on electron movement in a vacuum at a velocity exceeding the velocity of light there and concluded about the directional property of electron radiation [3]. However the advent of the relativity theory did Sommerfeld´s conclusions out of question, for the relativity theory keeps physical bodies away from the movement at a velocity equal or, the more so, exceeding the velocity of light. This conflict with the relativity theory might give a reason why the problem of supraluminal speed movement of charged particles was left out of consideration.