This collection provides codes that may be used to explore the theory of Frozen Waves (FWs). If you are unfamiliar with FWs, see Refs. [1] and [2] for an initial overview.
The folder fw contains C codes for computation and Wolfram Mathematica codes for plotting the field intensity of scalar Frozen Waves in lossy or lossless media. See Refs. [1], [2], [3], [4].
The folder fw-wm contains a purely Wolfram Mathematica version of the folder fw, i.e., Wolfram Mathematica codes for computing and plotting the field intensity of scalar Frozen Waves in lossy or lossless media. See Refs. [1], [2], [3], [4].
The folder fw2d-restricted contains C codes for computation and Wolfram Mathematica codes for plotting the field intensity of scalar restricted Frozen Waves 2D in lossy or lossless media, aiming for a 2D step function as the morphological function F. Similar codes may have been employed in Ref. [5].
The folder fw2d-restricted-wm contains a purely Wolfram Mathematica version of the folder fw2d-restricted, i.e., Wolfram Mathematica codes for computing and plotting the field intensity of scalar restricted Frozen Waves 2D in lossy or lossless media, aiming for a 2D step function as the morphological function F. Similar codes may have been employed in Ref. [5].
The folder fw2d-restricted contains C codes for computation and Wolfram Mathematica codes for plotting the field intensity of scalar restricted Frozen Waves 2D in lossy or lossless media, aiming for an image/picture as the morphological function F. Similar codes may have been employed in Refs. [5], [6] and [7] and [8].
This folder contains C++ codes for computation and Wolfram Mathematica codes for plotting the field intensity of a scalar surface FW (light sheet or 2D FW) apodized by gaussian profiles. Similar codes may have been employed in Ref. [9].
This folder contains C++ codes for computation and Wolfram Mathematica codes for plotting the field intensity of a scalar surface FW (light sheet or 2D FW) propagating through stratified media. Similar codes may have been employed in Refs. [10] and [11].
This folder contains C++ codes for computation and Wolfram Mathematica codes for plotting the field intensity of a scalar surface FW (light sheet or 2D FW) propagating through 4f systems. Similar codes may have been employed in Ref. [12].
The codes and routines were mainly developed and are updated by Jhonas O. de Sarro (@jodesarro).
The author is very grateful for the collaborations with Professor Leonardo A. Ambrosio of Applied Electromagnetics Group (AEG) from University of São Paulo (USP).
This project is protected under MIT License.
The codes in each folder may contain third-party codes with their own licenses. Please refer to the "license.txt" file in each folder.
[1] M. Zamboni-Rached, "Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves," Optics Express, vol. 12, no. 17, pp. 4001–4006, Aug. 2004, doi: 10.1364/OPEX.12.004001.
[2] M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, "Theory of 'frozen waves': modeling the shape of stationary wave fields," Journal of the Optical Society of America A, vol. 22, no. 11, pp. 2465–2475, Nov. 2005, doi: 10.1364/JOSAA.22.002465.
[3] M. Zamboni-Rached, "Diffraction-Attenuation resistant beams in absorbing media," Optics Express, vol. 14, no. 5, pp. 1804–1809, Mar. 2006, doi: 10.1364/OE.14.001804.
[4] M. Zamboni-Rached and M. Mojahedi, "Shaping finite-energy diffraction- and attenuation-resistant beams through Bessel-Gauss–beam superposition," Physical Review A, vol. 92, no. 4, p. 043839, Oct. 2015, doi: 10.1103/PhysRevA.92.043839.
[5] J. O. de Sarro and L. A. Ambrosio, "Constructing Millimeter-structured Surface Beams from Nondiffracting Zeroth-order Bessel Beams in Lossless Media," in 2019 PhotonIcs & Electromagnetics Research Symposium - Spring (PIERS-Spring), Rome, Italy: IEEE, Mar. 2020, pp. 283–288, doi: 10.1109/PIERS-Spring46901.2019.9017377.
[6] L. A. Ambrosio, "Millimeter-structured nondiffracting surface beams," Journal of the Optical Society of America B, vol. 36, no. 3, pp. 638–645, Feb. 2019, doi: 10.1364/JOSAB.36.000638.
[7] J. O. de Sarro and L. A. Ambrosio, "Surface beams resistant to diffraction and attenuation and structured at the millimeter scale," Journal of the Optical Society of America B, vol. 38, no. 3, pp. 677–684, Mar. 2021, doi: 10.1364/JOSAB.412756.
[8] A. H. Dorrah et al., "Light sheets for continuous-depth holography and three-dimensional volumetric displays," Nature Photonics, vol. 17, pp. 427–434, Apr. 2023, doi: 10.1038/s41566-023-01188-y.
[9] J. O. de Sarro, V. S. de Angelis, and L. A. Ambrosio, "Effects of Gaussian Apodization on the Propagation of Two-Dimensional Discrete Frozen Waves in Homogeneous Media," in 2023 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference (IMOC), Castelldefels, Spain: IEEE, Jan. 2024, pp. 286–288, doi: 10.1109/IMOC57131.2023.10379774.
[10] J. O. de Sarro and L. A. Ambrosio, "Propagation of ideal discrete 'frozen wave'-type light-sheets in lossless stratified media," Optics & Laser Technology, vol. 175, p. 110745, Aug. 2024, doi: 10.1016/j.optlastec.2024.110745.
[11] J. O. de Sarro and L. A. Ambrosio, "Two-dimensional Discrete Frozen Waves of Infinite Energy in Lossy Stratified Media," in 2024 IEEE Photonics Conference (IPC), Rome, Italy: IEEE, Dec. 2024, p. P35, doi: 10.1109/IPC60965.2024.10799710.
[12] J. O. de Sarro, M. Zamboni-Rached, G. Gouesbet, and L. A. Ambrosio, "Engineering the longitudinal intensity profile of optical beams after an arbitrary number of 4f-systems for light scattering applications," in 2025 Photonics & Electromagnetics Research Symposium - Spring (PIERS-Spring), Abu Dhabi, United Arab Emirates: IEEE, Dec. 2025, pp. 1–6, doi: 10.1109/PIERS-Spring66516.2025.11276750.