The invention of lasers has led to a boom of laser technology in various science and engineering of many interdisciplinary fields, which have been widely applied in optical manipulation, precision measurement, optical communication, laser processing, microscopy imaging, and so on. However, the simple propagation characteristics of the traditional Gaussian laser mode have hit the bottleneck in the further development of laser technology and failed to meet the ever-increasing needs in the related fields. Consequently, light field manipulation has emerged. By modulating the amplitude, phase, and polarization of the light field, many new types of spatially structured fields with novel physics effects or propagation properties have been proposed. Optical vortices and cylindrical vector beams are among the most well-known examples. To resist beam diffraction and environment disturbance, researchers have discovered a new class of spatially-structured field named “non-diffracting beam”. Theoretically, a non-diffracting beam can maintain its transverse intensity profile during propagation and can propagate over long distances with little beam spreading. Subsequently, a series of non-diffracting beams with different spatial structures, such as Mathieu beams, Weber beams, Airy beams, and Bessel beams, have been proposed, all of which exhibit the common characteristics of non-diffracting beams. As a typical non-diffracting beam, the Bessel beam quickly attracts great attention of research after being proposed as a propagation-invariant solution of the Helmholtz equation. Extensive research has been conducted on non-diffracting beams, including the applications in improving microscopy imaging quality and the performance of optical trapping. With the continuous development of light field manipulation, researchers have gradually attempted to control Bessel beams through different means by modulating or superposing the Bessel beams. Modulated Bessel beams can behave the self-accelerating propagation with nonlinear trajectories, with tunable on-axis intensities and polarizations, or even generate propagation-varied modes, which are different from traditional Bessel beams. Combining Bessel beams with other spatially structured light fields or optical elements for light field modulation would further expand the freedom degree for controlling Bessel beams. In this review, hence, we introduce the basic theory and generation methods of Bessel beams and review the research progress of the propagation control of Bessel beams, including the trajectory control, on-axis intensity management, longitudinally control of polarization, and self-similar Bessel beams.

In section 2, we introduce the basic characteristics of Bessel beams. Subsection 2.1 presents the solution of Bessel modes from the Helmholtz equation. Typical intensity and phase profiles of Bessel beams with different orders are represented (Fig. 1). The principle and several methods of generating Bessel beams are introduced in subsection 2.2. Fig. 2(a) shows the conical wave vector of Bessel beams, based on which two typical methods of generating Bessel beams with annular aperture and axicon are represented in Figs. 2(b) and 2(c), respectively. Based on the axicon-type phase, we propose a computer-generated hologram [Fig. 2(d)] and dielectric metasurface [Fig. 2(e)] to generate Bessel beams. Additionally, we introduce some other methods with the Fabry-Perot resonator [Fig. 2(f)] and optical fibers. Fig. 3 demonstrates the self-healing properties of Bessel beams. In section 3, we introduce the propagation trajectory control of Bessel beams. Subsection 3.1 presents the spiral Bessel beams, including radially self-accelerating beams (Fig. 4), accelerating rotating beams produced by the superposition of nonlinear vortex beams (Fig. 5), and spiraling zero-order Bessel beams produced by splicing the beam cone (Fig. 6). In subsection 3.2, the Bessel-like beams propagating along arbitrary trajectories based on caustic principle and pure phase modulation are shown in Figs. 7 and 8, respectively. The controllable spin Hall effect of the Bessel beam realized by geometric phase elements is introduced (Fig. 9). Subsection 3.3 represents the nonparaxial self-accelerating beams in Fig. 10, based on which tightly autofocusing beam is proposed (Fig. 11). In section 4, we introduce the axial intensity engineering of Bessel beam. In subsection 4.1, we introduce the theory of “Frozen Waves” (Fig. 12) and the modified “Frozen waves” following spiral and snake-like trajectories (Fig. 13). In subsection 4.2, the on-axis intensity modulation based on the spatial spectrum engineering [Fig. 14(a)] is introduced. By using metasurface, the on-axis intensity with rectangle and sinusoidal profiles are realized [Figs. 14(b) and 14(c)]. The on-axis intensity management is also applied to the self-accelerating Bessel beams to realize the on-demand tailored intensity along arbitrary trajectories [Fig. 14(d)]. Section 5 introduces the longitudinal control of the polarization of the Bessel beam. In subsection 5.1, the Bessel beams with propagation-varied polarization state are proposed based on the transverse- longitudinal mapping (Fig. 15). Based on the mapping, the vector Bessel-Gauss beams with propagation-variant polarization state and the corresponding self-healing are introduced (Fig. 16). In subsection 5.2, polarization oscillating beams constructed by superposing the copropagating optical frozen waves are shown in Fig. 17. In subsection 5.3, we introduce the longitudinal polarization control via the Gouy phases of beams. The self-accelerated optical activity in free space according to the Gouy phase difference between the Bessel beam and Laguerre-Gauss beam is shown in Fig. 18. We introduce the analogous optical activity in free space using a single Pancharatnam-Berry phase element, which can highly resemble the on-axis circular birefringence of beams. Fig. 19(a) shows the polarization rotator based on the theory. In addition, the off-axis circular birefringence, triggered by a tilted input Bessel beam, can generate the photonic spin Hall effect, which can be enhanced by inputting a self-accelerating Bessel-like beam [Fig. 19(b)]. In section 6, we introduce the self-similar Bessel-like beam, including self-similar beams with different scaling factors by solving the paraxial wave equation (Fig. 20), self-similar arbitrary-order Bessel-like beams based on the Fresnel integral (Fig. 21), and constructing arbitrary self-similar Bessel-like beams via transverse-longitudinal mapping (Fig. 22).

Modulated Bessel beams exhibit increased controllability during propagation while retaining the non-diffracting and self-healing properties. The trajectory, intensity, polarization state, and beam width can be flexibly controlled in the propagation direction. These characteristics have significant potential in various applications, including optical manipulation, microscopy imaging, and precision machining.