The new mechanism of plasma self-organization in transparent dielectrics with wide bandgap exposed to the intense tightly focused laser radiation was revealed which causes the generation of 3D periodic ring structures with subwavelength period both along the laser pulse propagation and in the radial direction. The mechanism involves formation of dense plasma burst in the pre-focal region that provides efficient scattering of the incident wave. The interference of a plane incident laser wave in the focal region and a divergent reflected one will form the standing wave pattern with local minima and maxima of laser field both in the direction of the incident wave propagation and perpendicular to it producing the ring patterns of effective ionization regions in the dielectric volume. Analytical and numerical simulations of the process of laser wave scattering on a near-spherical plasma object with dimensions both smaller and larger than the laser radiation wavelength are performed to verify the proposed model.