Frozen Waves: A curated collection of Frozen Wave codes

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], [2], [3], and [4] for an initial overview.

Folders and their content

📁 1-millimeter-structured-surface-beams

This folder contains Wolfram Mathematica codes for plotting the field intensity of a scalar surface FW (light sheet or 2D FW). Similar codes may have been employed in Ref. [5].

📁 2-resistant-surface-frozen-waves

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) resistant or not to the attenuation of a lossy media. Similar codes may have been employed in Ref. [6].

📁 3-gaussian-apodized-frozen-waves

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. [7].

📁 4-frozen-waves-in-stratified-media

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. [8] and [9].

📁 5-frozen-waves-and-4f-systems

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 Ref. [10].

Change log

Refer to the CHANGELOG.md file for the latest updates.

Authorship

The codes and routines were mainly developed and are updated by Jhonas O. de Sarro (@jodesarro).

Acknowledgements

The author is very grateful for the collaborations with Professor Leonardo A. Ambrosio of Applied Electromagnetics Group (AEG) from University of São Paulo (USP).

Licensing

This project is protected under MIT License. The codes may have their own licenses.

References

[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] 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.

[4] 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.

[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] 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.

[7] 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.

[8] 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.

[9] 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.

[10] 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.