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An explicit formula for a new type of beams, which in this work are calledthe "special" hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, usingthe method of the Hankel transform formulated in our previous work. Thefundamental properties of these beams are analyzed. The parameters that definethe beam shape have been identified and related to those of the fundamentalGaussian beam. The analytical expressions for the SHBG beams include anadditional parameter $\gamma$, which allows the beam's shape to be modified tosome extent. In the plane perpendicular to the propagation direction, thesebeams exhibit the annular nature. Interestingly, initially (i.e. near thebeam's spot) a single ring splits into a number of rings as one is moving alongthe beam. This is especially apparent for $\gamma$ close to unity, as thiseffect then appears for values of $z$ relatively small compared to the Rayleighlength i.e., where the energy concentration in the beam is still high. Thephase of the wave, whose behavior is in certain aspects typical of modes havingthe vortex character, is also studied in this paper.

Properties of "special'' hyperbolic Bessel-Gaussian optical beams