Results and Discussions In cylindrical coordinates, the Lommel function can be mathematically expanded as the superposition of Bessel functions, and the expression of PLBs can be derived by the Fourier transform of the Lommel function. We can readily learn from the expression of PLBs that the ring radius of PLBs is independent of the order, which is consistent with the characteristics of perfect vortex beams. Then, we theoretically simulate several kinds of PLBs with different parameters (Fig. 1). Finally, PLBs are experimentally generated. By the Lohmann-type detour phase coding method, the amplitude and phase of the complex wavefront of Lommel beams are coded simultaneously. The algorithm steps for constructing a binary CGH are given in Fig. 2, and on the basis of the algorithm, the binary CGH for generating Lommel beams is constructed (Fig. 3). The binary CGH of Lommel beams with other parameters can also be easily obtained by the same method proposed in this paper. We machine binary CGHs into binary amplitude masks by the holographic direct-writing printing system. The calculated CGHs are of 35000 pixel×35000 pixel, with the size of a pixel of 318 nm×318 nm, and hence the size of machined binary masks is 11.0 mm×11.0 mm. We construct an experimental setup to generate PLBs by using the machined binary amplitude masks (Fig. 4). The +1 order diffraction of masks is filtered by a keyhole, and then by the Fourier transform of Lommel beams, the PLBs are obtained and recorded by a scientific CCD camera (Flare 4M180-CL) (Fig. 5). It can be observed that the generated PLBs are in good agreement with their theoretical results. In addition, the orbital angular momentum of the PLBs is studied analytically, and the relationship between orbital angular momentum and mode of the fifth-order PLBs is plotted (Fig. 6).

Non-diffractive beams have been widely researched since their birth. The Lommel beams are a kind of nondiffractive beams, which have a complex structure and can be described by Lommel functions. The optical morphology of the non-diffractive Lommel beams can be modulated by three parameters, i.e., the topological charge n, the asymmetry parameter c₀, and the rotation angle ?₀. Apparently, Lommel beams differ from the one-parameter non-diffractive beams (Bessel beams, vortex beams, and Airy beams) and two-parameter non-diffractive beams (Mathieu beams and parabolic beams). The structure of three-parameter Lommel beams is more complex, and their optical morphology is more abundant than that of the one-parameter and two-parameter non-diffractive beams. For a traditional vortex beam, its structure is a bright ring around the middle dark core, and its topological charge affects the size of the bright ring radius. To solve this problem, researchers introduce the perfect vortex beam. The main optical property of the perfect vortex beam is that it has a ring vortex structure with stable size, namely that the size of the ring is independent of the topological charge. At present, perfect vortex beams mainly include classical perfect vortex beams and perfect elliptical vortex beams. This study attempts to produce perfect beams with more abundant optical morphology. In other words, we hope to generate perfect Lommel beams (PLBs) on the basis of diffraction-free Lommel beams, and the optical morphology of the produced PLBs can be adjusted by the three parameters at the same time.

Classical perfect beams are generated through the Fourier transform of Bessel beams. In this paper, we use the Fourier transform of Lommel beams to generate a new kind of perfect beams, i.e., PLBs. Complex amplitude modulation, namely that the amplitude and phase of beams are modulated simultaneously, is necessary for the generation of Lommel beams with a complex structure. It is easy to construct the amplitude modulation and phase modulation elements separately for beam generation, but the accurate alignment of the two elements is difficult. To produce high-quality Lommel beams, we need to introduce an encoding method to construct the complex amplitude modulation element, where the main purpose of encoding is to encode the amplitude and phase information of wavefront in one modulation element. Generally speaking, amplitude modulation is relatively easy. We adopt the Lohmann-type detour phase encoding method to modulate the complex amplitude of beams, which uses the diffraction effect of irregular grating, and by changing the grid spacing of local grating, we can obtain the required phase information at a certain diffraction level. With this encoding method, we construct a binary computer-generated hologram (CGH) that can produce Lommel beams. In the hologram, we can realize the amplitude modulation of beams by opening a rectangular optical aperture in the sampling unit of the hologram. Moreover, we can also realize phase modulation of beams by changing the two structural parameters of the aperture, i.e., the area of the aperture and the distance between its center and the sampling center. Then, the obtained binary CGHs for generating Lommel beams are machined into a mask with high resolution and high pixel number by the homemade holographic direct-writing printing system. For mask machining, first, the designed photolithography file (i.e., hologram) is automatically divided into a series of unit patterns with 600 pixel×600 pixel. These patterns are automatically input into a digital mirror device in accordance with their sequences and are scanned line by line for projection exposure on a Tianjin-III silver halide dry plate. When the lithography is completed, the silver halide dry plate is processed to obtain the amplitude mask. Finally, a high-quality Lommel beam is generated by the machined mask. On this basis, PLBs can be obtained by the Fourier transform of the generated Lommel beams.

We introduce and generate a type of new perfect vortex beams, i.e.,PLBs. Firstly, the theoretical mechanisms of PLB generation are deduced. Then, the experimental generation system is constructed to generate PLBs. The experiment system is mainly divided into two parts. The first part is to generate high-quality Lommel beams by the Lohmann-type detour phase encoding method, and the second part is to generate PLBs by the Fourier transform of the generated Lommel beams. The ring radius of the generated PLBs is not dependent on the topological charge value, and the optical distribution of PLBs can be controlled by three parameters, namely, the order, modulus of asymmetric parameters, and angle. This means that PLBs are perfect vortex beams with three degrees of freedom.