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. 2020 Jul 17;11:3597. doi: 10.1038/s41467-020-17439-3

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 3 — (Top row) Straight line segment. a Desired curve. b, c Experimentally measured transverse intensity and phase. d Experimentally obtained intensity volume over one Rayleigh length. (Second row) Superposition of building blocks allows creating cuspoid letters ‘AZ’ in propagation-invariant light. (Third row) Smart beam design facilitates complex structures: cardioid (i), nephroid (j), parabola (k), cusp (l) and the word ‘LIGHT’ in propagation-invariant light fields. n Proof of invariance during propagation at the example of the cardioid (i) over a distance of one Rayleigh length z_e.