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Volume 11
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Article

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

by
Adrián Garmendía-Martínez
1,*,
Francisco M. Muñoz-Pérez
1,2,
Walter D. Furlan
3,
Fernando Giménez
4,
Juan C. Castro-Palacio
1,
Juan A. Monsoriu
1 and
Vicente Ferrando
1
1
Centro de Tecnologías Físicas, Universitat Politècnica de València, 46022 València, Spain
2
Laboratorio de Fibra Óptica, División de Posgrado, Universidad Politècnica de Tulancingo, Tulancingo 43629, Hidalgo, Mexico
3
Departamento de Óptica, Universitat de València, 46100 València, Spain
4
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2023, 11(4), 946; https://doi.org/10.3390/math11040946
Submission received: 19 January 2023 / Revised: 2 February 2023 / Accepted: 9 February 2023 / Published: 13 February 2023
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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Abstract

In this work, we present a comparative analysis of different numerical methods to obtain the focusing properties of the zone plates based on Fibonacci and Cantor sequences. The Fresnel approximation was solved numerically in order to obtain the axial irradiance provided by these diffractive lenses. Two different methods were applied. The first one is based on numerical integration, specifically the Simpson integration method and the two-dimensional Gaussian quadrature. The second consisted in the implementation of the Fast Fourier Transform in both one and two dimensions. The axial irradiance of the lenses, the relative error with respect to the analytical solution, and the calculation time required by each method are analyzed and compared. From this analysis it was concluded that the Gauss method presents the best balance between accuracy and computation time. This analysis could be useful to decide the most convenient numerical method to be used for the study of more complex diffractive structures.
Keywords: numerical integration methods; fast fourier transform; fresnel integral; diffractive lenses numerical integration methods; fast fourier transform; fresnel integral; diffractive lenses

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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

APA Style

Garmendía-Martínez, A., Muñoz-Pérez, F. M., Furlan, W. D., Giménez, F., Castro-Palacio, J. C., Monsoriu, J. A., & Ferrando, V. (2023). Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses. Mathematics, 11(4), 946. https://doi.org/10.3390/math11040946

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