Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses
<p>Schematic diagram of Fresnel diffraction for a zone plate with transmittance function <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 2
<p>Zone plates based on (<b>a</b>) the Fibonacci sequence of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> and (<b>b</b>) the Triadic Cantor sequence of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Axial irradiance distribution computed through numerical and FFT methods, both one-dimensional and two-dimensional, for (<b>a</b>) a Fibonacci lens of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> and (<b>b</b>) a triadic Cantor lens of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>. These calculations have been carried out with a number of sampling points of <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>Analysis of the numerical error provided by the used methods to calculate the axial irradiance for different values of sampling points for (<b>a</b>) the Fibonacci lens of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> and (<b>b</b>) the Triadic Cantor lens of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>Computation time for different numbers of sampling points, for the calculation of the irradiance distribution of the (<b>a</b>) Fibonacci lens of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math> and (<b>b</b>) Triadic Cantor lens of order <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p> ">
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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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
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 StyleGarmendía-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