Skew-step function governing main and intermediate intensity spots in radial carpet beams

Abstract Radial carpet beams (RCBs) are produced by the diffraction of a plane wave from radial structures, resulting in unique properties such as self-healing, non-diffracting behavior, accelerating propagation, core-area amplifying, and discrete intensity patterns. While the mathematical formulation of RCBs is well-established, the precise behavior of their intermediate radial intensity spots as the number of grating spokes varies remains underexplored. In this study, we investigate the relationship between the number of grating spokes and the generated spots in the intermediate radial distances. Interestingly, the number of intermediate radial intensity peaks increases with the number of grating spokes, not in a smooth linear fashion but through a series of tilted steps, each slightly lower than the end of the previous one, showing a peculiar yet structured behavior. We reveal a pair of skew-step functions that govern this dependency, with both functions having the same form but being laterally shifted. This demonstrates an inherent order similar to systematic natural phenomena, such as Fibonacci-based phyllotaxis in plants, spiral shell formations, planetary spacing described by the Titius–Bode law, and digit distributions predicted by Benford’s law. This work provides new insights into the structured behavior of RCBs, enhancing our understanding of the underlying principles governing optical beam dynamics and their parallels to natural systems.