Simulation of diffraction patterns of annular apertures

Montilla-López, Yomber; Suárez-Carreño, Franyelit; Linares, Brexys; Rojas, Loguard

doi:10.1590/1806-9126-RBEF-2025-0346

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Articles • Rev. Bras. Ensino Fís. 47 • 2025 • https://doi.org/10.1590/1806-9126-RBEF-2025-0346 linkcopy

Simulation of diffraction patterns of annular apertures

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Abstract

In this paper, Fraunhofer diffraction patterns generated by square and circular annular apertures are studied. The annular region is defined as the difference between two concentric areas of equal geometry: an outer fixed aperture and an inner closed obstacle, whose size is varied by a scaling parameter $h$ . This parameter modifies the geometry of the aperture and, consequently, the shape and energy distribution in the diffraction pattern. Exact equations for intensity and analytical expressions for the fraction of energy contained in the region of maximum intensity are derived. For visualization and numerical analysis, codes are implemented in Maple 25, and a graphical user interface is designed in Python 3.13 that allows the evolution of diffraction patterns to be observed through animation as $h$ varies. The study developed may be useful in optical applications, particularly in imaging systems.

Keywords:
Fraunhofer diffraction; annular aperture; Maple simulation; Python interface; diffraction patterns

$h$	$I_{\max}^{1}$	$X_{\min}^{1}$	$I_{\max}^{2}$	$X_{\max}^{2}$	$X_{\min}^{2}$	$I_{\max}^{3}$	$X_{\max}^{3}$	$X_{\min}^{3}$
0.000	1.000	$\pm 3.142$	0.047	$\pm 4.493$	$\pm 6.283$	0.016	$\pm 7.725$	$\pm 9.425$
0.495	1.000	$\pm 2.643$	0.170	$\pm 4.261$	$\pm 6.296$	0.051	$\pm 7.863$	$\pm 9.931$
0.825	1.000	$\pm 2.219$	0.261	$\pm 3.712$	$\pm 5.369$	0.224	$\pm 7.009$	$\pm 8.717$

$h$	$I_{\max}^{1}$	$ρ_{\min}^{1}$	$I_{\max}^{2}$	$ρ_{\max}^{2}$	$ρ_{\min}^{2}$	$I_{\max}^{3}$	$ρ_{\max}^{3}$	$ρ_{\min}^{3}$
0.000	1.000	3.832	0.017	5.136	7.016	0.004	8.417	10.173
0.495	1.000	3.153	0.095	4.831	7.201	0.012	8.679	10.961
0.825	1.000	2.631	0.155	4.180	6.040	0.077	7.652	9.468

Parameter	Square		Circular
$h$	$Δ x_{\min}^{1}$	$F (Δ x_{\min}^{1})$	$Δ x_{\min}^{1}$	$F (Δ x_{\min}^{1})$
0.000	6.283	82%	7.663	84%
0.495	5.286	47%	6.306	48%
0.825	4.437	14%	5.263	15%

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[T1FN1] The notations $I_{\max}^{n}$ , $I_{\min}^{n}$ , $X_{\max}^{n}$ , and $X_{\min}^{n}$ represent the position of the optimal values corresponding to the $n$ -th maximum or minimum of relative intensity measured from the center. For $n = 1$ , the central maximum is identified, followed by the first minimum, until the sequence of local maxima and minima of the diffraction pattern is complete.