A Fully Phase‐Modulated Metasurface as An Energy‐Controllable Circular Polarization Router

Abstract Geometric metasurfaces primarily follow the physical mechanism of Pancharatnam–Berry (PB) phases, empowering wavefront control of cross‐polarized reflective/transmissive light components. However, inherently accompanying the cross‐polarized components, the copolarized output components have not been attempted in parallel in existing works. Here, a general method is proposed to construct phase‐modulated metasurfaces for implementing functionalities separately in co‐ and cross‐polarized output fields under circularly polarized (CP) incidence, which is impossible to achieve with solely a geometric phase. By introducing a propagation phase as an additional degree of freedom, the electromagnetic (EM) energy carried by co‐ and cross‐polarized transmitted fields can be fully phase‐modulated with independent wavefronts. Under one CP incidence, a metasurface for separate functionalities with controllable energy repartition is verified by simulations and proof‐of‐principle microwave experiments. A variety of applications can be readily expected in spin‐selective optics, spin‐Hall metasurfaces, and multitasked metasurfaces operating in both reflective and transmissive modes.

2 Text S1. Derivation process of Jones matrix for meta-atom In this part, the specific derivation process of Jones matrix is exhibited. Based on linear polarization basis, the Jones matrix with inherent linear responses along x-and y-directions and rotation characteristics is given as:  is defined as original phase, which is provided by propagation phase by varying the dimensions of meta-atoms, and φ'(x, y) = 2θ(x, y) is the additional degree of freedom, which is provided by geometric phase from rotating meta-atoms and only works in the cross-polarized field. It is noted that the response of φ'(x, y) under orthogonal CP incidence are conjugate symmetry, attributed to the inherent property of geometric phase.

Text S2. Design principle of meta-atom
Referring to equivalent circuit of filters in microwave region, it is theoretically possible to achieve unity transmittance and full 2π phase coverage by involving high order LC resonance in a band-pass filter model, which would provide more degree of freedom for phase tuning.
Here, the basic microwave meta-atom is composed of five metallic layers and four dielectric substrate layers as presented in Figure S1 For the sake of demonstration, one meta-atom is selected to present the EM response in microwave region. The simulated magnitude and phase profiles of transmission spectra under linearly polarized incidence and CP incidence are depicted in Figure S1(b)-S1(c), respectively. It can be observed that under the illumination of x-and y-polarized incidences, the amplitude of both txx and tyy is higher than 0.9 and the phase difference ∆φ = φxxφyy is kept to π/2 within 9.5 GHz -10.5 GHz frequency range, guaranteeing the effective generation and conversion of transmitted wave with LHCP and RHCP states.
Meanwhile, the amplitude of co-polarized and cross-polarized transmitted coefficient under LHCP incidence can reach 0.7 around the center frequency of 10 GHz as shown in Figures. S1(d)-S1(e), which implies that the transmission coefficient of energy carried by co-polarized and cross-polarized can simultaneously approach 0.5 separately ( 22 co cross tt ). The phase difference between co-and crosspolarized coefficients (∆φcir = |φco -φcross|) is also fixed to π/2, providing a phase criterion to independently control orthogonal CP output channels in transmitted field. According to the design criterion above, a library of 24 meta-atoms, endowing uniform phase interval of π/12 in order to obtain the full phase coverage, is established with further parametric optimization. The simulated circularly 6 polarized phase delays (φco and φcross) and corresponding phase difference are exhibited in Figure S1(f), offering the basic effective microwave elements for full manipulation of transmitted wave. Furthermore, some basic electromagnetic characteristics of the proposed meta-atom is illustrated in this part. In microwave region, metal can be regarded as perfect electric conductor and the loss tangent of dielectric materials is usually in the order of 10 -3 . Thus, the system is supposed as a lossless system in the analyzations and simulations. Since the meta-atom proposed in this paper is symmetric in structure 7 and totally passive (without any active component), all of the proposed meta-atoms are reciprocal. It means that there exist S12 xy = S21 yx , S12 yx = S21 xy in the general scattering matrix, where the first (second) subscript presents for the region of incident (scattered) field, and the first (second) superscript shows the polarization state of incident (scattered) wave. Additionally, the reciprocity is verified by simulations as shown in Figure S2, where the scattering responses of transmission coefficients are exhibited. Figure S2(a) and S2(b) show the amplitude and phase responses of meta-atom with px = 4.8 mm, py = 5.7 mm. It can be seen that the transmission curves S12 xy and S21 yx of the meta-atom are in coincident, while S12 yx and S21 xy are in same tendency, which successfully verify the reciprocity of proposed structure. For the matching property, theoretical model is supposed to be matched with free space which is convenient for calculating the phase-modulation process. However, since there is inevitable reflection within the design process of meta-atom structures, the practical system would be mismatched. During the construction of metasurfaces, the phase responses of meta-atoms have to approach discrete spatial phase distribution, for example required by deflection or OAM generation. To achieve the phase response as accurate as possible, there would be some optimization process of the meta-atom's phase response and the amplitude response would inevitably be sacrificed. So, equivalent network of the meta-atom is considered mismatched with free space. The prototypes proposed in this paper are fabricated by using classical printed circuit board (PCB) technique, and which is simply introduced as follows. First, the polyfluortetraethylene dielectric substrates with double copper-cladding layers are prepared. Then the copper surface treatment is conducted for the dry film adhesion purpose, which is used as sensitive material to record printed circuit structures by exposing to ultraviolent (UV) rays. After exposure process, the non-sensitizing dry film needs to be rinsed by sodium carbonate photoresist developer. Furthermore, the exposed copper  The schematic representation of the near-field measurement system is exhibited in Figure S4.
Measurements are conducted by a setup surrounded by absorbers in order to minimize parasitic reflections. A 2 GHz -18 GHz dual-polarized wideband horn antenna is used as the feeding source to launch the circularly polarized quasi-plane wave (LHCP and RHCP), and placed at a distance d > 20λ0 away from the meta-devices. A fiber optic active antenna is used as field probe to measure both the amplitude and the phase of the electric field. The probe is fixed on two translation stages controlled by a motion controller and its position is incremented by a step of 2 mm. Both the horn antenna and the field probe are connected to an Agilent 8722ES vector network analyzer, which is adopted to measure the complex S11 and S21 parameters including both amplitude and phase. Here S11 and S21 parameters represent the reflection and transmission coefficients of the system under test, respectively. The probe is oriented in two directions in order to measure the two components x E and y E (horizontal and vertical) of the transmitted electric field, and then the CP transmitted field at one fixed pixel can be calculated by co x y E E i E for co-pol component and cross x y E E i E for cross-polarized component under LHCP incidence, including both amplitude and phase. With the variation of the position of the field probe via the motion controller, the xoy and xoz planes can be totally covered, and the experimental near-field intensities and phase profiles can be measured. Figure S4. Schematic illustration of the measurement setup for near-field mapping.

Text S5. Scheme demonstration in optical regime
Here, the basic optical meta-atom shown in Figure S5(a) is composed of five layers of noble metal film and four silicon dioxide spacers, realizing a 4 th -order LC band-pass filter as shown in Figure S5(b).
A representative meta-atom is selected to present the response of the equivalent broadband filter model.
The simulated magnitude and phase profiles of transmission spectra under linearly polarized incidence are depicted in Figure S5 and width (py) of the patch are shown in Figure S5(e) under the illumination of LHCP incidence.
According to the above variation rule and design criterion, a library of 8 meta-atoms endowing uniform phase interval of π/4 in order to obtain the full 2π-phase range coverage, is established with further parametric optimization. The simulated circularly polarized phase delays φco and φcross are displayed in Figure S5(f), offering basic effective elements for full manipulation of transmitted wave.
In order to validate the proposed formalism that can be expanded in optical regime, we conduct a metasurface working with λ0 (opt) = 1550 nm, which can generate converging beam in co-polarized transmitted field and vortex beam carrying OAM with mode l = 1 in cross-polarized transmitted field (same wavefronts with metadevice-1 in main text) under the spin-up illumination. It can be observed from Figure S6(a) and S6(b) that, under the impinging of spin-up σillumination, the co-polarized output wavefront is focus beam with focal length f(opt) = 5λ0 (opt), and the vortex beam with OAM l = 1 is generated in the cross-polarized field. When the incident light state changed into spin-down σ + , the copolarized output shown in Figure S6(c) performs focusing beam with focal length f(opt) = 5λ0 (opt), which is totally similar to co-polarized output in Figure S6(a). Meanwhile, the cross-polarized output exhibits converged vortex beam with OAM l = -1. The converged vortex energy intensity along propagation direction (focal length is about 1.6λ0 (opt)) is shown in Figure S6(d), and the corresponding vortex ringshape energy distribution and helical phase pattern are exhibited in the inset of Figure S6(d). The

11
optical results indicate that the proposed phase scheme for simultaneous manipulation of both output channels can be effectively expanded into other frequency regime.
Here it should be noticed that the meta-atom is not necessarily to be multi-layered structure. Other metallic or dielectric nano-structures can be appropriate for the fabrication of optical metasurfaces, as long as the two phase-modulation degrees (propagation phase and geometric phase) can be satisfied by tailoring some specific parameters of meta-atom, for example the dielectric pillars shown in 10.1103/PhysRevLett.118.113901.  13

Text S6. Evaluation of efficiency of the proposed metadevices
The efficiency of meta-device is evaluated and discussed in this part, and two efficiency characteristics are defined here, including wavefront utilization efficiency (UE) and transmission efficiency (TE). The energy of the required field can be obtained by the sum of measured intensity at each point in the detecting plane, which is collected from the probe and analyzed in the vector network analyzer as detailed in Text S4. The incident energy is obtained by the same process but without placing the fabricated sample. Different from the measured energy intensity, which is obtained by graphing the energy values point-by-point in the detection plane, the efficiency against frequency is acquired by integrating all the energy values on the whole detection plane. Here, the transmission efficiency can be calculated by: where Pout and Eout expresses the total transmitted energy and electric field, Pin and Ein express the incident energy and electric field, and s is the area of the measured plane. The measured TE of fabricated metadevice-1 and metadevice-2 with Δφ = 90° are 77% and 82% at the center frequency 10 GHz, respectively. These results prove the transmission abilities of the designed metadevices and guarantee the high performances of the required functionalities.
Furthermore, the utilization efficiency can be analyzed by: where Pcir is the total energy with circular polarizations (both LHCP and RHCP) in the transmitted field, representing the sum of both co-polarized and cross-polarized components. It can be seen in inset of Figure 4h in the main text that the wavefront utilization efficiency for all the metadevices are approaching 100% within the measured frequency range of 9 GHz -11 GHz, which indicates that almost all energy in the transmission can be modulated to perform corresponding preset functionalities. Only a little part of energy is lost at several frequency points due to the uncompleted CP conversion of several meta-atoms. Indeed, some meta-atoms do not achieve the exactly required ∆φ between x E and y E , resulting in the generation of undesired elliptically polarized transmitted components.