Polyadic Cantor Fractals: Characterization, Generation, and Application as Ultrasonic Lenses

Abstract

The term fractal was coined in 1975 by Benoit Mandelbrot. Since then, fractal structures have been widely used by the international scientific community. Its range of applications includes multiple areas, such as optics, physics, cryptography, medicine, economics, and so on. The application of fractal structures to modulate light beams in the field of optics has been extensively studied, and it has been shown that in some cases these new fractal lenses improve the response of traditional lenses. Fractal lenses are able to provide beamforming capabilities, and allow the optimization of the optical beam according to the specific requirements. In some applications, it may be necessary to improve the focus in a certain area, while in others it may be critical to obtain a sharp attenuation by means of destructive interference. It may even be required a beam profile with multiple focus and a certain control over them. This work investigates the application of fractal structures based on Polyadic Cantor sets as ultrasonic lenses, analyzing how the relation between the different design parameters and the performance of the lens. It is shown that the working frequency becomes a precise control mechanism that can modify dynamically the focus position of the lens.

Numerical simulation and laboratory measurements on the influence of fractal dimension on the acoustic beam modulation of a Polyadic Cantor Fractal lenses

“…It can also be noted that for the diffraction phenomenon to occur, sizes comparable to the wavelength, λ , must be considered. It can be observed in Fig.1a a fractal at stage 2 with central gap (g c ) value of g c = 0.4(1 − 4r), where r is the scaling factor [17]. The gap between blocks of elements when the stage is increased is called ε.…”

Polyadic Cantor Fractal Ultrasonic Lenses: Design and Characterization

“…Among the possible geometries that can be analyzed, fractal structures have already been proposed [7,8]. Particularly, Generalized Polyadic Cantor Sets (GPCSs) are very attractive because they can be generated with simple algorithms, and this is the main reason why the majority of studies are based on these type of fractals [9,10].…”