Fig. 1. Optical manipulation based on metasurface. (a) Optical manipulation by metasurface can be described by a 2×2 Jones matrix, which establishes the relationship between the properties of incident and output lights; (b) realization of different optical functionalities by metasurface with different degrees of freedom (DOFs) in Jones matrix

Fig. 2. Structure design and applications of Jones matrix metasurface with one degree of freedom. Unit design of Jones matrix with one degree of freedom (a) by geometric phase based on the rotational angle of nanorod, and (b) (c) by changing the structure size, including the length long and short axes of the nanorod or the radius of the disk, to achieve resonance (or propagation) phase or amplitude modulation; optical functionalities of (d) metalens^[35], (e) high efficiency hologram^[45], and (f) anomalous refraction^[52] based on geometric phase; optical functionalities of (g) anomalous refraction^[67] and (h) hologram^[73] based on resonance or propagation phase; full-color printing based on amplitude modulation by (i) changing the length of nanorod^[82] and (j) (k) the radius of disk^{[84, 90]}

Fig. 3. Structure design and applications of Jones matrix metasurface with two degrees of freedom. (a)(b) Realization of resonance (or propagation) phase control and amplitude control in two orthogonal directions based on the size changing of the length of long and short axes of the nanorod; (c) realization of Jones matrix metasurface with two degrees of freedom based on the size change of the length of long axis of the nanorod and the rotational angle; (d) amplitude and phase control of one component of Jones matrix based on the rotation-angle freedom of two nanorods; (e) polarization controlled dual holographic images^[91] based on unit structure in Fig. 3(a); (f) stereoscopic 3D image display^[92] based on the unit structure of Fig. 3(b); (g) independent control of the anomalous refraction angles of the left and right circular polarizations^[93] based on the unit structure in Fig. 3(c); (h) integration of color printing and holographic image^[32] based on the unit structure in Fig. 3(d)

Fig. 4. Structure design and applications of Jones matrix metasurface with three degrees of freedom. (a) Realization of three-degree-of-freedom Jones matrix based on two independent transmission phases along the major and minor axes and rotational angle of the nanorod; (b) independent phase control of arbitrary orthogonal polarization states^[114]; (c) independent amplitude control of arbitrary orthogonal polarization state^[115]; (d) independent holographic image under the combination of three incidence-output polarization states based on the independent phase control of three components of Jones matrix^[117]

Fig. 5. Structure design and applications of Jones matrix metasurface with four degrees of freedom. (a) Schematic of nanorod with rotational-angle degree of freedom and transmission-phase degree of freedom along one direction; (b) diagram of generating optical field with arbitrary amplitude, phase, and polarization, which does not meet the geometric phase constraints; (c) diagram of generating arbitrary optical field which meets the geometric phase constraints; (d) arbitrary optical field generation to realize vector beam with arbitrary patterns and polarization states^[118] based on single nanorod structure; (e)–(h) other structures based on the nanorod in Fig. 5(a) to realize arbitrary optical field, where the detour phase is introduced in Fig. 5(h); (i) arbitrary optical field generation to realize multi-channel and multi-angle optical control^[119] based on the unit structure in Fig. 5(e); (j) arbitrary optical field generation^[120] based on the unit structure in Fig. 5(f); (k) arbitrary optical field generation^[120] based on the unit structure in Fig. 5(g); (l) arbitrary optical field generation to realize perfect vector vortex beam^[121] based on the unit structure in Fig. 5(h); (m) conversion between two arbitrary amplitude, phase, and polarization states^[122] based on the unit structure in Fig. 5(h)

Fig. 6. Structure design and applications of Jones matrix metasurface with six degrees of freedom. (a) Design of unit structure to realize Jones matrix with six degrees of freedom using two nanorod structures, each with three complete degrees of freedom, including two transmission phases along the major and minor axes and the rotation-angle degree of freedom; (b) design of unit structure to realize Jones matrix with six degrees of freedom using four nanorod structures, which only considering its rotation-angle degrees of freedom and position-dependent detour phase; (c) continuous polarization change in the propagation direction^[126] based on the unit structure of Fig. 6(a); (d) independent amplitude and phase control of arbitrary orthogonal polarization state^[128] based on the unit structure in Fig. 6(a); (e) conversion of any polarization state into the same polarization state^[129] based on the unit structure in Fig. 6(a); (f) independent printing and holographic image integration under the combination of three different incidence-output polarization states (xx, yy,and xy/yx)^[127] based on the unit structure in Fig. 6(b)

Fig. 7. Structure design and applications of Jones matrix metasurface with eight degrees of freedom. (a) Realization of Jones matrix with eight degrees of freedom by double-layer compact nanorod structure^[131]; (b) an application example of producing a 3D spiral pattern with radial polarization under RCP incidence and two different holographic images on different transverse planes with gradually varying polarization states^[131] based on the structure in Fig. 7(a); (c) realization of Jones matrix with eight degrees of freedom by two layer metasurfaces^[130], where the distance between the two layers is hundreds of microns; (d) experimental verification of the Jones matrix with eight degrees of freedom^[130] proposed in Fig. 7(b), where independent printings and holographic images are observed under the four combinations of incident and detection polarization (xx, xy, yx, yy); (e) independent amplitude and phase control on any set of two polarizations and conversions to arbitrary two polarizations^[130] based on the Jones matrix metasurface with eight degrees of freedom