# An Ultrasonic Lens Design Based on Prefractal Structures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling and Simulation

_{i}, which defines the pattern for plane wave incidence and wavelength λ for a FZP is given by

_{max}= 2, the total number of rings will be M = 8; thus, the total number of zones are 2M + 1 = 19. The scaling ratio between successive stages of growth, γ, is given by

_{gaps}is the number of desired gaps, and D is the fractal dimension. In this work, n

_{gaps}= 3 and D = 9/10. For the first stage (s = 0), the bar length is L, and this length varies with its stage through the expression ${L}_{\gamma}^{s}$. On the other hand, the coefficients ${b}_{{c}_{j}^{i}}$ in Equation (2) depends on γ and ε (the lacunarity) [16].

_{o}(IPW) impinges on the axysimetric lenses upward along the y direction.

_{incident}is the lens incident pressure.

## 3. Results and Discussion

^{3}and a sound velocity of 1500 m/s. The solved problem has 0.88 × 10

^{6}elements.

_{incident}is the pressure of the incident wave. For the FZP lens, the focal gain value was 19.4 dB, while for FRZP it was 13.1 dB. These results revealed that the focusing effect of FRZP lens do not improve the FZP lens. Therefore, the redistributing scattering centers (solid rings) of the fractal lens was modified by moving the distance W/2 of the dispersing element towards the rotation axis (see Figure 3), a

_{i}being now the distance from the axis of rotation to the center of the ring segment i. This new structure was referred to as the “modified FRZP”. Figure 5 shows a comparison of the transverse section of the sound pressure level along the y-axes of the three lens considered. It can be seen that only by redistributing the solid rings could the acoustic focalizing behavior improve considerably. For the modified FRZP, the focal gain was G

_{focus}= 20.9 dB. Furthermore, a remarkable feature can be observed: Fractal structures produce multiple foci along the transversal axes with a high sound pressure level. This feature is characteristic of fractal diffractive lenses, which is a result of the phase sampling inherent to these type of lenses [17].

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic section diagram of the generation for a Fresnel zone plate (FZP) and FZP considered.

**Figure 2.**Schematic section diagram of the generation for a fractal zone plate (FRZP) from stage 0 to stage 2.

**Figure 4.**Schematic diagram of the configuration simulated in the numerical domain where the solutions are obtained.

**Figure 5.**Transverse section of sound pressure level along the y-axis for the FZP (blue), FRZP (red) and modified FRZP (black) lenses.

**Figure 6.**Spatial distribution of the sound pressure level (dB) for (

**a**) the modified FRZP; (

**b**) the FZP studied here.

**Figure 7.**Transverse section of absolute pressure field along the x-axis at the focus for (

**a**) the Fresnel zone plate (FZP) and (

**b**) the fractal zone plate (FRZP) studied here. The insets show the radial pressure fields distribution.

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**MDPI and ACS Style**

Castiñeira-Ibáñez, S.; Tarrazó-Serrano, D.; Rubio, C.; Candelas, P.; Uris, A.
An Ultrasonic Lens Design Based on Prefractal Structures. *Symmetry* **2016**, *8*, 28.
https://doi.org/10.3390/sym8040028

**AMA Style**

Castiñeira-Ibáñez S, Tarrazó-Serrano D, Rubio C, Candelas P, Uris A.
An Ultrasonic Lens Design Based on Prefractal Structures. *Symmetry*. 2016; 8(4):28.
https://doi.org/10.3390/sym8040028

**Chicago/Turabian Style**

Castiñeira-Ibáñez, Sergio, Daniel Tarrazó-Serrano, Constanza Rubio, Pilar Candelas, and Antonio Uris.
2016. "An Ultrasonic Lens Design Based on Prefractal Structures" *Symmetry* 8, no. 4: 28.
https://doi.org/10.3390/sym8040028