The nonlinear Schrödinger equation (NLSE), one of the most fundamental partial differential equations in nonlinear science, plays a central role in characterizing the dynamic behavior of complex nonlinear physical systems. Mathematically, it is considered a form of nonlinear parabolic partial differential equation. In optics, it serves as a fundamental theoretical framework for analyzing nonlinear optical field properties and related dynamical phenomena. Spatial solitons, owing to their self-preserving spatial structure during propagation, have emerged as a central topic in nonlinear optics since their theoretical prediction, attracting extensive research interest. In nonlinear media, diffraction and self-induced nonlinear effects can be balanced, leading to stable propagation of optical beams. In mathematics, the rose curve is named for its petal-like geometrical appearance. Its light intensity distribution exhibits a petal-shaped rotationally symmetric profile arising from the coupling between the angular cosine modulation and radial distribution functions, which exemplifies the trigonometric geometrical characteristics inherent to the polar coordinate system. However, current research on rose-shaped beams, particularly in nonlinear media, remains relatively limited, and their potential for advanced light-field modulation in nonlinear systems has not yet been fully explored.
This paper presents, for the first time, the concept of rose curve modulated beams (RCMBs) and further extends the theoretical framework by introducing radial and angular modulation parameters to construct radial-angular composite rose curve beams (RACRBs) with enhanced structural flexibility. This model not only inherits the unique geometric characteristics of conventional rose curve beams but also significantly enriches the spatial structure of the optical field through the coupling of multiple modulation parameters, providing a novel theoretical framework for the design of complex structured light fields. Based on the nonlocal nonlinear Schrödinger equation in conjunction with the Huygens–Fresnel integral, an analytical expression describing the propagation of rose curve beams in strongly nonlocal nonlinear media (SNNM) is derived, establishing a solid theoretical foundation for investigating their propagation dynamics. Numerical simulations are performed to systematically analyze the evolution of the beam width, intensity distribution, and critical power. The results demonstrate that the spatial intensity profile at the source plane can be precisely tailored by appropriately adjusting the radial modulation parameter, angular modulation parameter, and rotation angle, thereby enabling the generation of a variety of beam configurations with distinct spatial characteristics. When the RACRBs parameters are set to (0, n), the beam evolves into a stable spatial soliton with a tunable petal number and spatial arrangement. The petal morphology can be continuously manipulated through parameter variation, highlighting the high degree of structural controllability and design flexibility of the proposed beam.
When both the radial modulation parameter m and the angular modulation parameter n are nonzero, indicating the simultaneous presence of radial and angular modulation, the proposed beam exhibits propagation behaviors that are absent in conventional structured beams. Numerical results reveal the emergence of a pronounced intensity inversion phenomenon, leading to the formation of multiple stable intensity inversion nodes during propagation. The positions of these nodes evolve periodically with the propagation distance. Furthermore, variations in the optical power effectively modulate the locations of the inversion nodes while preserving the overall structural stability of the optical field. This distinctive behavior represents a key characteristic of rose curve beams and clearly distinguishes them from conventional Gaussian, Airy, and Bessel beams. In addition, the introduction of a rotation angle causes the inversion nodes to rotate synchronously about the beam propagation axis while maintaining the integrity of the beam profile, thereby enabling coordinated control of both the beam structure and the inversion nodes. The initial incident power of the beam affects the refractive index of the medium, thereby influencing the evolution period; higher initial power results in a shorter evolution cycle. Numerical simulations confirm that rosette beams exhibit stable long-distance propagation in strongly nonlocal media. These findings provide valuable insights for future research on rosette beams and demonstrate their potential applications in optical communication and optical particle manipulation.