## 2 Design Method of the Off-Axis Pseudo Bessel Beam and Multiple Pseudo Bessel Beams  
A metasurface has a controllable phase shift to realize the control of the electromagnetic wave by adjusting the unit cells. As shown in Figure 1a, the metasurface on the xoy-plane can be used to generate pseudo Bessel beams, propagating in the directions of \(u_{k}\) or \(u_{l}\), which means any direction in the coordinate system. It is important to calculate the compensation phase of each unit cell whose coordinate is \((x_{ij},y_{ij},0)\), and the unit cell we proposed has the structure of two rings and one cross, which will be explained in detail in the next section. The compensation phase that the unit cell provides is controlled by the size of the unit cell, and the compensation phase distribution of the metasurface is decided by the beams we prepare to generate. As the generation of a pseudo Bessel beam needs the coherent superimposition of both inward conical waves and outward conical waves [13; 33], it is necessary to modulate these waves by the metasurface. As shown in Figure 1a, the energy radiated from the feed horn to the metasurface can be modulated by the unit cells to form the inward conical wave. The inward conical wave is shown as dashed lines in Figure 1a,b; it is worth pointing out that all of the unit cells contribute for the inward conical wave so that they can form a conical surface. After passing through the propagation axis, the inward conical wave becomes the outward conical wave, and a pseudo Bessel beam will be formed in the overlapping area of the two. The wave vector \(k\) of the inward conical wave can be divided into the transverse component \(k_{t}\) and the longitudinal component \(k_{l}\), i.e., \(k=\sqrt{k_{t}^{2}+k_{l}^{2}}=2\pi/\lambda\) (\(\lambda\) is the wavelength in free space). It is easy to find that if the transverse components of the inward and outward conical waves have the same magnitude and opposite directions, a standing wave will be formed in the transverse direction. In other words, the angle \(\alpha\) between the inward conical wave and the propagation axis (\(\tan\alpha=k_{t}/k_{l}\)) remains the same in this condition, and this will prohibit the energy to leak out in the transverse direction. Meanwhile, the longitudinal components of the inward and outward conical waves will have the same magnitude and direction, so a traveling wave can be formed in the longitudinal direction. Therefore, in the superimposed region, a pseudo Bessel beam can propagate along the propagation axis with all of the electromagnetic energy. From the mentioned generation procedure of a pseudo Bessel beam, we can know that forming a standing wave in the transverse direction is the key that all of the energy can propagate along the desired direction. This also means the included angle \(\alpha\) between the inward conical wave and the propagation axis needs to remain the same, and as the focused beams do not meet this requirement, they cannot remain uniform in energy distribution during a long distance [26; 27; 28; 29]. Only in this way can the \"standing wave\" condition be satisfied, and all of the energy can be propagated along the propagation axis without leaking to other directions. Based on the above principle and discussion, we can give a method for generating an off-axis pseudo Bessel beam using a metasurface. As it is shown in Figure 1b, the first thing we need to do is to determine the propagation direction of the pseudo Bessel beam; the unit vector of the pseudo Bessel beam can be expressed as \(\hat{u}_{k}=(\sin\theta_{k}\cos\varphi_{k},\sin\theta_{k}\sin\varphi_{k},\cos \theta_{k})\), where \(\theta_{k}\) and \(\varphi_{k}\) are the pitch angle and the azimuth angle of the desired beam in the coordinate system, respectively. Then we need to determine the ratio of the wave vector's transverse component to the longitudinal component; that is, determine the angle \(\alpha\).