# Design of Acoustic Bifocal Lenses Using a Fourier-Based Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Relation between ZP Transmittance Function and Focusing Profile

## 3. Application of the Fast Fourier Transform to the Lens Binary Sequence

## 4. Fourier-Based Design Algorithm

- Step 1: Initial Calculations

- Step 2: Generation of the focusing profile sequence $F\left({u}_{n}\right)$

- Step 3: Generation of the target transmittance sequence $t\left(n\right)$

- Step 4: Generation of the design binary sequence $b\left(n\right)$

## 5. Simulation Results and Discussion

- Step 1: The first step consists of performing the necessary calculations to obtain the ${u}_{1}$ and ${u}_{2}$ parameters, using Equations (12)–(15).$$\begin{array}{ccc}\hfill \lambda & =& 0.15\phantom{\rule{4.pt}{0ex}}\mathrm{mm}\hfill \\ \hfill {u}_{0}& =& 20\hfill \\ \hfill {z}_{0}& =& 480\phantom{\rule{4.pt}{0ex}}\mathrm{mm}\hfill \\ \hfill a& =& 53.7\phantom{\rule{4.pt}{0ex}}\mathrm{mm}\hfill \\ \hfill {u}_{1}& =& 24\hfill \\ \hfill {u}_{2}& =& 16\hfill \end{array}$$As ${u}_{1}$ and ${u}_{2}$ are integer numbers, the first step is already finished;
- Step 2: The focusing profile sequence, $F\left({u}_{n}\right)$, is generated using parameters N, ${u}_{1}$ and ${u}_{2}$. Figure 4a depicts the target focusing profile, where the two foci can be sharply distinguished;
- Step 3: Applying the IFFT to the focusing profile sequence, the transmittance sequence, $t\left(n\right)$, is generated as shown in Figure 4b;
- Step 4: Figure 4c depicts $b\left(n\right)$, the result of binarizing the transmittance sequence. With this final step, the design process is completed. In this particular case, the generated binary sequence is $b\left(n\right)=\left\{1011010110101101011010110101101011010110\right\}$.

**Figure 4.**Example of the design of a bifocal ZP with foci locations at $z1=400$ mm and $z2=600$ mm. (

**a**) Focusing profile sequence. (

**b**) Transmittance sequence. (

**c**) ZP binary sequence. (

**d**) ZP bifocal normalized focusing profile. Inset: Bifocal ZP layout with transparent areas (red) and phase-reversal areas (blue).

- Step 1: Calculations to obtain ${u}_{1}$ and ${u}_{2}$, using Equations (12)–(15).$$\begin{array}{ccc}\hfill \lambda & =& 0.15\phantom{\rule{4.pt}{0ex}}\mathrm{mm}\hfill \\ \hfill {u}_{0}& =& 20\hfill \\ \hfill {z}_{0}& =& 420\phantom{\rule{4.pt}{0ex}}\mathrm{mm}\hfill \\ \hfill a& =& 50.2\phantom{\rule{4.pt}{0ex}}\mathrm{mm}\hfill \\ \hfill {u}_{1}& =& 28\hfill \\ \hfill {u}_{2}& =& 12\hfill \end{array}$$As ${u}_{1}$ and ${u}_{2}$ are integer numbers, the first step is completed;
- Step 2: The focusing profile sequence, $F\left({u}_{n}\right)$, is generated using parameters N, ${u}_{1}$ and ${u}_{2}$. Figure 12a depicts the target focusing profile, where the two foci can be sharply distinguished;
- Step 3: Applying the IFFT to the focusing profile sequence, the transmittance sequence, $t\left(n\right)$ is generated as shown in Figure 12b;
- Step 4: Figure 12c shows the result of binarizing the transmittance sequence, which ends the design process. In this design example, the generated binary sequence is $b\left(n\right)=\left\{1001101100100110110010011011001001101100\right\}$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Fourier transforms on binary sequences for a Fresnel ZP (

**a**,

**c**,

**e**,

**g**) and a Fibonacci ZP (

**b**,

**d**,

**f**,

**h**). Binary sequences: (

**a**,

**b**). Focusing profile sequences: (

**c**,

**d**). Normalized focusing profile as a function of u: (

**e**,

**f**). Normalized focusing profile as a function of z: (

**g**,

**h**). The $F\left(n\right)$ sequence is computed as the FFT of the $b\left(n\right)$ sequence. Focusing profiles are computed using the Rayleigh-Sommerfeld diffraction integral.

**Figure 3.**Generation of the ZP binary sequence for a monofocal focusing profile (

**a**,

**c**,

**e**,

**g**) and a bifocal focusing profile (

**b**,

**d**,

**f**,

**h**). Target focusing profiles: (

**a**,

**b**). Transmittance sequences: (

**c**,

**d**). Binary sequences: (

**e**,

**f**). Normalized focusing profiles, computed with the Rayleigh-Sommerfeld diffraction integral, as a function of u: (

**g**,

**h**).

**Figure 5.**Normalized focusing profiles vs. axial coordinate for the conventional Fresnel ZP (blue), bifocal ZP (red) and no lens (yellow) cases. Insets depict the ZP layouts for the Fresnel ZP (top) and bifocal ZP (bottom).

**Figure 6.**Bifocal ZP focusing profiles vs. axial coordinate. Red dashed line: $N=40$. Blue solid line: (

**a**) $N=20$ and (

**b**) $N=80$. Insets depict the ZP layouts for $N=20$ (top) and $N=80$ (bottom). (

**c**) FLHM at the first (blue) and second (red) focus against N.

**Figure 7.**Bifocal ZP focusing profiles vs. axial coordinate. Blue solid line (

**a**) ${u}_{1}=24$ and ${u}_{2}=18$ (inwards shift case), (

**b**) ${u}_{1}=26$ and ${u}_{2}=16$ (outwards shift case). Dashed red line corresponds to the initial design example with ${u}_{1}=25$ and ${u}_{2}=17$. Insets depict the ZP layouts for the focusing profiles depicted in blue.

**Figure 8.**(

**a**) Bifocal ZP focusing profiles vs. axial coordinate. Blue solid line: resolution limit case with ${u}_{1}=22$ and ${u}_{2}=20$. Dashed red line: initial design example with ${u}_{1}=25$ and ${u}_{2}=17$. Inset depict the ZP layout for the resolution limit case. (

**b**) Resolution limit against N.

**Figure 9.**Bifocal ZP focusing profiles vs. axial coordinate for different design frequencies. Blue solid line: (

**a**) $f=5$ MHz, (

**b**) $f=1$ MHz, (

**c**) $f=500$ kHz, (

**d**) $f=250$ kHz. Dashed red line corresponds to the initial design example with $f=10$ MHz.

**Figure 10.**Bifocal ZP focusing profiles vs. axial coordinate for different operating frequencies. Blue solid line: (

**a**) $f=9$ MHz, (

**b**) $f=9.5$ MHz, (

**c**) $f=10.5$ MHz, (

**d**) $f=11$ MHz. The dashed red line corresponds to an operating frequency equal to the design frequency ($f=10$ MHz).

**Figure 11.**(

**a**) Foci location against operating frequency. (

**b**) Foci FLHM against operating frequency. Blue and red lines refer to the first and second focus at the bifocal focusing profile, respectively.

**Figure 12.**Example of the design of a bifocal ZP with foci locations at $z1=300$ mm and $z2=700$ mm. (

**a**) Focusing profile sequence. (

**b**) Transmittance sequence. (

**c**) ZP binary sequence. (

**d**) ZP bifocal normalized focusing profile. Inset: Bifocal ZP layout.

**Table 1.**FLHM values for Figure 6a,b.

N | ${\mathbf{FLHM}}_{\mathbf{1}}$ [mm] | ${\mathbf{FLHM}}_{\mathbf{2}}$ [mm] | ${\mathbf{FLHM}}_{\mathbf{2}}/{\mathbf{FLHM}}_{\mathbf{1}}$ |
---|---|---|---|

20 | 29.43 | 67.27 | 2.29 |

40 | 15.62 | 33.63 | 2.15 |

80 | 7.81 | 17.42 | 2.23 |

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**MDPI and ACS Style**

Fuster, J.M.; Pérez-López, S.; Candelas, P.
Design of Acoustic Bifocal Lenses Using a Fourier-Based Algorithm. *Sensors* **2021**, *21*, 8285.
https://doi.org/10.3390/s21248285

**AMA Style**

Fuster JM, Pérez-López S, Candelas P.
Design of Acoustic Bifocal Lenses Using a Fourier-Based Algorithm. *Sensors*. 2021; 21(24):8285.
https://doi.org/10.3390/s21248285

**Chicago/Turabian Style**

Fuster, José Miguel, Sergio Pérez-López, and Pilar Candelas.
2021. "Design of Acoustic Bifocal Lenses Using a Fourier-Based Algorithm" *Sensors* 21, no. 24: 8285.
https://doi.org/10.3390/s21248285