A Novel Multiband Fractal Antenna Based on Three-Dimensional Mandelbrot-Set Structure

Mandelbrot fractals are a class of geometric structures characterized by non-integer dimensions, whose unique quasi-self-similarity provides fractal antennas with enhanced electrical lengths and diverse electromagnetic properties. This work presents the systematic investigation into the application of the 3D Mandelbrot set in antenna design for the first time. Through theoretical analysis, structural generation and numerical simulations, we explore the effects of varying iteration orders and power exponents on antenna performance. The results show that the “bud-like” features on the fractal surface significantly extend the antenna’s electrical length, enabling both fundamental and higher-order resonances. At lower frequencies, the antenna exhibits pronounced high-gain radiation patterns. The iteration order can be adjusted to tune the operational bandwidth, while changes in the power exponent offer favorable multiband characteristics. Further, characteristic mode analysis reveals that the 3D fractal structure supports fewer and more orthogonal modes at low frequencies, whereas at higher frequencies, it displays multi-modal behavior with correspondingly more complex far-field radiation patterns. A prototype fabricated via metal 3D printing for a second-order, eighth-power configuration validates the design, exhibiting a wide impedance bandwidth from 2.72–16.13 GHz, a peak measured gain of 7.8 dBi, and radiation efficiency up to 70%. The proposed design demonstrates excellent broadband and multi-band radiation capabilities, showing potential for future multi-frequency communication systems.