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9), possesses infinite energy. However, finite-energy NDWs can be constructed by concentrating the spectrum Anm (kz , ω) in the surrounding of a straight line of the type ω = Vkz + bm instead of collapsing it exactly over that line [12, 63]. In such a case, the NDWs get a finite energy, but, as we know, are endowed with finite field depths, that is, they maintain their spatial forms for long (but not infinite) distances. 9, does represent ideal NDWs, it is difficult to use it for obtaining analytical solutions, especially when there is the task of eliminating the backward components.

6 of [1]. It may be useful to repeat that a Bessel beam is characterized by an ‘‘extended focus’’ along its propagation axis, so that its energy cannot be concentrated inside a small region in the transverse plane: It needs, indeed, to be reconstructed continuously by the energy associated with the ‘‘lateral rings’’ (evolving along closing conical surfaces), which constitute its transverse structure. 1 A General Introduction that is, is constructed so as to concentrate its energy within a spot that becomes very small at a certain point of its propagation axis, and afterward diffracts rapidly.

Let us stress that, as such a choice is not unique, many alternative sets of values L and qR exist, which also yield excellent results. 47: it represents, with high fidelity, the Bessel beam truncated at z = 0. 8b. One can see that the result really corresponds to a Bessel beam truncated by a finite aperture. 5 Subluminal NDWs (or Bullets) Let us now obtain in a simple way non-diffracting subluminal pulses, always as exact analytic solutions to the wave equations [19]. We shall adopt in this section a less formal language (perhaps more intuitive or more physical), and we shall confine it to ideal solutions, but such solutions will be constructed for arbitrarily chosen frequencies and bandwidths, once more avoiding any recourse to the non-causal (backward-moving) components.

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Conformal Invariants by Thomas J. M.

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