# Polyadic Cantor Fractal Ultrasonic Lenses: Design and Characterization
## 2 Polyadic Cantor Fractal Lens Generation  
Figure 1 shows a Polyadic Cantor Fractal Lens (PCFL) across the XY-coordinate plane with plane wave excitation. The acoustic field is modulated by the lens and originates a focus on a perpendicular axis to the lens. The distance between the position of the focus and the lens plane is known as the focal distance (FL). In some applications, it may be necessary to have multiple foci. The PCFL is built using the GPCS algorithm. First, the initial element (stage \(s=0\)), known as the initiator, is substituted by a set of \(N\) segments of identical lengths \(W_{0}=Lr\). The rparameter is known as the scaling factor and should take values from 0-1/2 in order to produce a GPCS; see Figure 2a. These segments are separated by gaps. If \(N\) is even, there is a central gap of width \(g_{c}\) and \(N-2\) lateral gaps of width \(\varepsilon\); see Figure 2b. If \(N\) is odd, the central gap is split into two halves of width \(g_{c}/2\) at the sides of the central segment, and there are \(N-3\) lateral gaps of width \(\varepsilon\); see Figure 2c. As can be seen in Figure 2, the number of total gaps is always given by \(n_{gaps}=N-1\). Figure 1: Polyadic Cantor Fractal Lens (PCFL) built with concentric rings. The lens focus is located in an axis orthogonal to the lens plane at distance FL. Figure 2: (**a**) GPCS main parameters, (**b**) even GPCS and (**c**) odd GPCS. This way, the stage \(s=1\) of the GPCS, known as the generator, is constructed. Both the initiator (\(s=0\)) and generator (\(s=1\)) have exactly the same length. This procedure is repeated as many times as required in the following stages. The number, distribution and widths of the segments of the generator affect the homogeneity and texture of the fractal. To characterize these features, parameters such as the fractal dimension (\(D\)), the number of elements (\(N\)), the fractal stage s, the central gap width \(g_{c}\) and the lateral gap width \(\varepsilon\) are considered; see Figure 2a. The fractal dimension is defined as,  
\[D=-\frac{\ln(n_{gaps}+1)}{\ln r} \tag{1}\]  
with \(n_{gaps}\) the total number of gaps of the generator and \(r\) the scaling factor. If \(N\) is fixed, \(n_{gaps}\), as well, and \(r\) must be modified to obtain different values for the fractal dimensions. Therefore, the fractal dimension relates directly to the transparency of the fractal. An alternative way to refer to the fractal gap distribution is through a parameter known as the central gap fraction \(f_{gc}\). This parameter ranges from 0-1 and gives an indication of the weight of the central gap over the total gap space in the generator. The width of the central gap can be related to the central gap fraction using,  
\[g_{c}=L(1-rN)f_{gc} \tag{2}\]  
As mentioned above, the even-order GPCS has a unique central gap of width \(g_{c}\), while the odd-order GPCS splits the central gap into two gaps of widths \(g_{c}\)/2 at both sides of the central segment. The lateral gap widths of the generator can be calculated with the following expressions,  
\[\varepsilon= \begin{cases}\frac{L(1-Nr)-g_{c}}{N-2}&N\geq 4\text{ if }N\text{ is even}\\ \\ \frac{L(1-Nr)-g_{c}}{N-3}&N\geq 5\text{ if }N\text{ is odd}\end{cases}  \tag{3}\]  
As stated previously, this is a recursive procedure. In each stage, the different segments are substituted by generator copies of the same length. Thus, as the stage evolves, the number of segments increases, while the segment length decreases. At stage s, the segment lengths can be calculated with,  
\[W_{s}=Lr^{s} \tag{4}\]  
Although this iterative procedure can be repeated indefinitely, once the second stage (\(s=2\)) is achieved, the iterative procedure is truncated due to mechanical limitations with the construction of the device. Therefore, the final structure is not a real fractal, but a prefractal. The PFCL is then generated from the prefractal structure.