Objective Structured light beams with controllable intensity distribution have received great interest recently. Many new-type beams exhibit unique physical properties including diffraction-free, self-bending, orbital-angular-momentum-carrying, and abruptly autofocusing. Particularly, non-diffracting beams can retain their intensities and shapes during propagation and have been widely used in laser micromachining, particle manipulation, and microscopic imaging. The first type of non-diffracting beam is the Bessel beam, which was first discovered by Durnin in 1987. It is a wave packet solution expressed as a Bessel function and derived from the Helmholtz equation in a cylindrical coordinate system. Additionally, Bessel beams have a doughnut-shaped intensity distribution and phase singularity at their center and also exhibit many unique optical properties. Most non-diffracting beams are associated with the known, exact solutions of the wave equation and match with special forms of the angular spectrum. It is the angular spectrum that determines the beam propagation dynamics, as well as the intensity profiles. To achieve diverse transverse shapes of non-diffracting beams, a traditional method is to solve the wave equation under different coordinates. For example, the cosine beam, the Mathieu beam, and the Weber beam were obtained by solving a wave equation in a rectangular coordinate system, an elliptic coordinate system, and a parabolic coordinate system, respectively. Later, more non-diffracting beams were constructed by superimposing the basic fundamental non-diffracting beams. Recently, the typical accelerating Airy beam also exhibits unique non-diffracting characteristics as well as self-acceleration and self-healing. However, current methods are limited to producing only a limited non-diffracting structured light beam, which greatly restricts their applications. Consequently, it is imperative to create a greater variety of non-diffracting beams with diverse transverse intensity distributions so that the capabilities of non-diffracting structured light beams can be enhanced. We demonstrate a universal method for designing and generating non-diffracting structured light beams with arbitrary shapes. Such light beams can be tailored by predefining appropriate spectral phases. Unlike conventional approaches, our method overcomes the traditional limitation that these non-diffracting beams are always constructed from wave equation solutions. The ability to produce non-diffracting beams with arbitrary transverse shapes offers potential benefits for manipulating particles along arbitrary transverse trajectories and could inspire new applications in optical micromachining, manipulation, and wavefront control. Methods We develop an efficient, simple, and universal optical caustic approach from the perspective of geometric optics. Concretely, our idea is to design a suitable spectral phase to produce a non-diffracting beam with the desired shape. Then, the relationship between the spectral phase and the beam structure is established based on the stationary phase approximation that relies on the cancellation of components due to rapid phase oscillation. Subsequently, the desired non-diffracting beams are generated by imposing the constructed spectral phase on a plane wave. Further, a modified algorithm is developed via spectral phase superposition. In this way, non-diffracting beams with arbitrary structures are generated. Experimentally, the non-diffracting beams with arbitrary shapes can be generated at the focal plane of a Fourier transform lens illuminated by a plane wave, as described in Fig. 4. Results and Discussions We break through the traditional constraints of solving the Helmholtz partial differential equation to obtain arbitrary non-diffracting beams. The constructed non-diffracting beams exhibit diverse transverse structures, such as circular and parabolic shapes (Fig. 2), kidney-shaped and 8-shaped (Fig. 3), as well as heart-shaped (Fig. 5). The maximum intensity is always located in the predefined trajectory because the second derivative for the phase part described by Eq. (6) equals zero. The one-to-one correspondence between the transverse beam shape and the spectral phase is established (Eq. (5) and Eq. (7)), and this makes our approach possible to manipulate and generate various non-diffracting beams. Moreover, an improved algorithm is developed to generate non-diffracting beams with arbitrary transverse shapes (Eq. (8)) that can consist of both convex and concave trajectories. The numerical results are consistent with the experimental results (Fig. 6). The non-diffracting property is also discussed, as demonstrated in our numerical and experimental results which exhibit an obvious non-diffracting property. Such non-diffracting beams would also be advantageous for manipulating particles following arbitrary transverse shapes. Conclusions In summary, we develop an effective strategy and a practical technique to achieve non-diffracting beams with an arbitrary transverse intensity distribution. The proposed approach breaks the limitation that the classical non-diffracting beams are always constructed from the solutions of wave equations. The beam shape can be readily customized by the desired spectral phase constructed using optical caustics. The constructed non-diffracting beams with arbitrary shapes greatly enrich the family of structured light beams and would be beneficial to manipulating particles following such transverse shapes. They are likely to provide ideas for new applications in optical micromachining, manipulation, and wavefront control.
