Theoretical and numerical analysis of the transmission function of the focusing system with high numerical aperture was conducted. The purpose of the study was to form a thin light tube in a focal area using the azimuthally polarized radiation. It was analytically shown that, due to destructive interference of two beams formed by two narrow rings, it is possible to overcome not only the full aperture diffraction limit but also the circular aperture limit. In this case, however, the intensity at the center of the focal plane is significantly reduced, which practically leads to the tube rupture. It was numerically shown that long thin one-piece tubes may be formed through the aperture apodization with diffractive axicon phase function or with complex transmission function of Laguerre-Gaussian or Airy-Gaussian beams. 1. Introduction Introducing a narrow annular aperture to the tightly focused cylindrical beams with radial or azimuthal polarization, the blocking light in almost all central parts of the lens [1–3] is a simple but energetically expensive way of forming long narrow beams in the focal region. In the case of radial polarization a thin thread of light is formed, while in the azimuthal polarization a light tube is formed. Moreover, the transverse dimension corresponds to the scalar diffraction limit. In other types of polarization the focal spot (or ring) is larger because of the contribution of the various components of the electromagnetic field to the focal region. To increase the efficiency and overcome the diffraction limit more sophisticated ways of full aperture apodization of function are used. It can be either a pure phase or an amplitude-phase distribution [4–9]. Thus, as a rule, reducing the focal spot size is accompanied by the redistribution of energy from the central part to the sidelobes. This situation is consistent with the Toraldo di Francia theory [10], according to which possible to obtain infinitely narrow central spot due to the growth of sidelobes. But this growth is sometimes several times [8, 9] or even orders [11] higher than the reducing of central light spot. The presence of significant sidelobes limits the use of “super-resolution” elements in representing systems and optical data recording, when the acceptable intensity level in the sidelobes is less than 30% with respect to the central peak [12]. However, the optimization procedures controlling the growth of the sidelobes lead to the inevitable broadening of the central spot size [12, 13]. It was shown in [14] that the introduction of the radial phase jump on radians
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