Here, we design the holographic metasurface in terms of the Fresnel holography technique to generate the structured light field. Suppose the target light field is *U*0(*x*0,*y*0), the diffraction light field at the propagation distance of *d* away from the target field can be expressed by Fresnel integration of *U*(*x*,*y*)=*∫∫U*0(*x*0,*y*0)*K*(*x*-*x*0,*y-y*0)d*x*0d*y*0, where the point spread function of *K*(*x*-*x*0,*y-y*0) satisfies *K*(*x*-*x*0,*y-y*0)= [exp(*jkd*)/(*jλd*)]exp{*jk*[(*x*-*x*0)2+(*y*-*y*0)2]/(2*d*)} with *k* = 2π/*λ* denoting the wave number of the incident field and *λ* representing the wavelength33. Since the scatterers of metasurface are distributed discretely, the diffraction field should be discretized, and it can be written as

Where *p*, *q*, *m* and *n* are integers, *δ**x* and *δ**y* represent the sampling intervals along the *x* and *y* axes of the diffraction field, and *δ**x*0 and *δ**y*0 represent the sampling intervals along the *x*0 and *y*0 axes of the target field. While the diffraction field performs the reverse Fresnel diffraction, the obtained field is *U**H*(*x*0´,*y*0´) = *U*0(*x*0,*y*0)/(*λd*)2 and the discretized form is *U**H*(*mδ**x*,*nδ**y*) = *U*0(*mδ**x*,*nδ**y*)(*λd*)2. One can see that the diffraction field is the same as the target field except for the constant term. This means that the target field can be reproduced at the defaulted distance, and it is also the working principle of metasurface holography. When the target field is a radially structured light including several concentric vortices encoded by different polarization states, one of polarization mode *U*0*i*(*mδ**x*0,*nδ**y*0) can be expressed as,

Where C*i* represents the amplitude of vortex, *r**i* is the radius of vortex, *w**i* denotes the waist radius of vortex, and *l**i* is the topological charge. For different polarization mode, these parameters may take the same or different values. The amplitude and phase information of the holographic metasurface can be obtained by inserting Eq. (2) into Eq. (1). For simplification, we set the amplitude threshold to construct the metasurface hologram. Here, the phase distribution of radially structured light is realized with the help of the phase delays introduced through rotating the L-shaped nanoholes. As we know, the phase delay introduced by an anisotropic nanohole equals to twice of the rotation angle of nanohole and it is always carried by the cross circularly polarized light31. Therefore, one spiral phase with the circularly polarized state can be easily generated through rotating the nanoholes.

As we know, one linear polarization state of cos*γ*e*x*+sin*γ*e*y* with the polarization angle of *γ* can be expressed as the superposition of two orthogonal circularly polarized states, namely, cos*γ*e*x*+sin*γ*e*y*= 20.5[exp(*jγ*)e*R*+exp(*-jγ*)e*L*]/2, where e*x*=(1, 0), e*y*=(0, 1), e*R*=20.5(1, *-j*)/2 and e*L*=20.5(1, *j*)/2 represent the unit vectors for *x*-polarization, *y*-polarization, right- and left-handed circular polarization. Thus, two suits of nanoholes with opposite rotation angles can consist of one metasurface with linear polarization output. Figure 1A shows the sketch map of one structured light generator with *x*-polarization output. The compound metasurface CM is formed through combining the complementary structures of R1 and R2, whose nanoholes are randomly chosen from two metasurfaces of M1 and M2 corresponding to the right- and left-handed circular polarization. This compound metasurface takes effect under the *x*-polarized light illumination.

Similarly, for the structured light encoded by two orthogonal linearly polarized states, one can construct the compound PESLG in terms of above principle. Suppose one OAM mode of exp(*jl*1*φ*) takes the *x*-polarization and the other OAM mode of exp(*jl*2*φ*) takes the *y*-polarization, the target field with two orthogonal linearly polarized OAM modes can be written as,

While *w*1 = *w*2 and *r*1 = *r*2, the above equation can be simplified into exp(*-r*2/*w*2)exp(*jpφ*) [exp(*jqφ*)e*x*+exp(*-jqφ*)e*y*] with *p*=(*l*1 + *l*2)/2 and *q*=(*l*1-*l*2)/2 and it is the so-called vector vortex beam34. While *p* = 0, it changes into the pure vector beam. Figure 1B shows the schematic diagram for the generation of PESL by the constructed holographic metasurface. This holographic metasurface is composed of four sets of rotated nanoholes, where two sets of nanoholes take effect under the right-handed circularly polarized light illumination and the other two sets of nanoholes take effect under the left-handed circularly polarized light illumination. The rotation angle *θ* of nanohole at any position equals to *l*1*ϕ*/2 for the first set, -*l*1*ϕ*/2 for the second set, *l*2*ϕ*/2 + π/4 for the third set and -*l*2*ϕ*/2 + π/4 for the fourth one.

In order to ensure the quality of the generated PESL, we should optimize the structure parameters of metasurface. The structure parameters include the thickness of silver film, the length and width of L-shaped nanohole and the separation of two adjacent nanoholes, which are labeled in the magnified nanohole inserted in the lower right corner of Fig. 1A. Here, the working wavelength of *λ* is set at 632.8nm and the propagation distance of *d* is set at 15µm. Through the optimization, the separations of nanoholes of *δ**x* and *δ**y* along two orthogonal directions takes 200nm, which satisfies Nyquist sampling theorem *δ*2*N* ≤ *λd* with the sampling number of *N* taking 200. Two lengths of *l**a* and *l**b* take 120nm and two width of *w**a* and *w**b* take 35nm. The thickness of silver film takes *h* = 150nm. The optimization process is finished with the help of finite-difference time-domain technique34.