# Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### Numerical Methods Used to Solve the Fresnel Integral

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ZP | Zone Plate |

FFT | Fast Fourier Transform |

DFT | Discrete Fourier Transform |

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**Figure 1.**Schematic diagram of Fresnel diffraction for a zone plate with transmittance function $t({x}_{0},{y}_{0})$.

**Figure 2.**Zone plates based on (

**a**) the Fibonacci sequence of order $S=8$ and (

**b**) the Triadic Cantor sequence of order $S=3$.

**Figure 3.**Axial irradiance distribution computed through numerical and FFT methods, both one-dimensional and two-dimensional, for (

**a**) a Fibonacci lens of order $S=8$ and (

**b**) a triadic Cantor lens of order $S=3$. These calculations have been carried out with a number of sampling points of $N=200$.

**Figure 4.**Analysis of the numerical error provided by the used methods to calculate the axial irradiance for different values of sampling points for (

**a**) the Fibonacci lens of order $S=8$ and (

**b**) the Triadic Cantor lens of order $S=3$.

**Figure 5.**Computation time for different numbers of sampling points, for the calculation of the irradiance distribution of the (

**a**) Fibonacci lens of order $S=8$ and (

**b**) Triadic Cantor lens of order $S=3$.

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**MDPI and ACS Style**

Garmendía-Martínez, A.; Muñoz-Pérez, F.M.; Furlan, W.D.; Giménez, F.; Castro-Palacio, J.C.; Monsoriu, J.A.; Ferrando, V.
Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses. *Mathematics* **2023**, *11*, 946.
https://doi.org/10.3390/math11040946

**AMA Style**

Garmendía-Martínez A, Muñoz-Pérez FM, Furlan WD, Giménez F, Castro-Palacio JC, Monsoriu JA, Ferrando V.
Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses. *Mathematics*. 2023; 11(4):946.
https://doi.org/10.3390/math11040946

**Chicago/Turabian Style**

Garmendía-Martínez, Adrián, Francisco M. Muñoz-Pérez, Walter D. Furlan, Fernando Giménez, Juan C. Castro-Palacio, Juan A. Monsoriu, and Vicente Ferrando.
2023. "Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses" *Mathematics* 11, no. 4: 946.
https://doi.org/10.3390/math11040946