Independent phase modulation for quadruplex polarization channels enabled by chirality-assisted geometric-phase metasurfaces

Geometric-phase metasurfaces, recently utilized for controlling wavefronts of circular polarized (CP) electromagnetic waves, are drastically limited to the cross-polarization modality. Combining geometric with propagation phase allows to further control the co-polarized output channel, nevertheless addressing only similar functionality on both co-polarized outputs for the two different CP incident beams. Here we introduce the concept of chirality-assisted phase as a degree of freedom, which could decouple the two co-polarized outputs, and thus be an alternative solution for designing arbitrary modulated-phase metasurfaces with distinct wavefront manipulation in all four CP output channels. Two metasurfaces are demonstrated with four arbitrary refraction wavefronts, and orbital angular momentum modes with four independent topological charge, showcasing complete and independent manipulation of all possible CP channels in transmission. This additional phase addressing mechanism will lead to new components, ranging from broadband achromatic devices to the multiplexing of wavefronts for application in reconfigurable-beam antenna and wireless communication systems.

S are also equal, which successfully verify the reciprocity of proposed structure. In fact, in our proposed scheme for full transmission phase manipulation, we just consider one propagation direction (from port 1 to port 2) of the electromagnetic wave.
2. Lossless. 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 can be supposed as a lossless system in the analyses and simulations. Practically, if the system is proposed with lossy media, the equivalent scattering matrix would not be unitary and the efficiency of the system would be further affected and reduced. Since our scheme only works on the phase modulation, the change of amplitude would not affect the application of the scheme and it is still effective.

3.
Matching. The ideal model is supposed to be matched to free space, which is convenient for calculating the phase-modulation process. However, since there is inevitable reflection within the design process of metasurface (or 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 for 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. On the other hand, the interior and exterior rotation in meta-atom structure would also introduce mismatching in the system, resulting in the increase of reflection components. As it can be observed from Supplementary Figure 2a, 2c and 2e, when the meta-atom is imposed with the interior and exterior rotations, the amplitude of S xx and S yy are gradually decreased. So, equivalent network of the meta-atom is considered mismatched to free space.

Supplementary Note 2: Analysis of chirality-assisted phase introduced by the proposed meta-atom
The chirality-assisted phase factor is introduced by interior rotation between adjacent patch layers of the meta-atom, which is simply achieved by rotating each single patch layer with different angles 1 In order to simplify the cascaded Jones matrix conveniently, here, the rotation angles of each patch layer is set as 1  =− , 2 0  = , and 3  = , and the cascaded transmission expression can be obtained as: Based on Eq. (S2), the Jones matrix of the meta-atom, which is set with the specific interior rotation and no exterior rotation, can be extracted and illustrated by T1: 3  2  i 2  i 4  3  2  xx  yy  xx  yy  xx  yy  xx  yy  xx  yy  xx  yy  3  2  3  xx  yy  xx  yy  xx  yy  xx  yy  xx In order to calculate the output phase patterns in four transmission coefficients in T1, here it is supposed that the linear transmission amplitude for each ideal birefringent wave plate model is regarded as 1 (|t0xx| = | t0yy| = 1). The phase difference and phase combination between the two linear orthogonal directions are defined as 0xx 0yy For the further simplification and calculation of Jones matrix of equivalent meta-atom with interior rotation and no exterior rotation, the following two relationships can be obtained: Eq. (S5) expresses the chirality-assisted phase functions of four CP channels transmission coefficients.
It can be seen that the phase patterns of two co-polarized channels L-L and R-R have been decoupled with different phase pattern by introducing interior rotation angle ϑ, while the cross-polarized output phase in channel L-R and R-L are still maintained in same state. It is indicated that the interior rotation angle ϑ would just produce decoupling influence in the co-polarized channels and would have no effect on the cross-polarized output fields. It should be noted that in this part, Eq. (S5) is established on the idealized assumption that the three patch layers are considered as perfectly independent ideal wave plates. However, practically in this work, the three patch layers are metallic elements in the equivalent filter circuit as analyzed in Supplementary Note 1, meaning that the coupling between each layer would unavoidably affect the output final phase. Therefore, the phase in four CP channels of each meta-atom for metadevices construction (as illustrated in Fig. 3b in main text), are finally fixed with full wave simulation and optimization processing, which can effectively counteract the influence produced by metallic-layer coupling.
Moreover, as for the proposed meta-atom imposed with both interior and exterior rotation, the equivalent Jones matrix is expressed as LL LR

 =  
, which is further calculated based on T1. It is known that the exterior rotation provides the geometric phase and the exterior rotation angle is set as θ.
The derivation process can be described as follows: is the rotation matrix. Eq. (S6) expresses the general form of Jones matrix. Four elements A, B, C, D are independent from each other and represent for different transmission characteristics in the four channels, which are described as: where t0xx and t0yy represent the linear transmission coefficients of birefringent single-layer patch, ϑ is the interior rotation angle provided by three independent patch layers, and θ shows the exterior angle introduced by rotating the whole meta-atom with all three patch layers.

Supplementary Note 3: Derivation of basic method for constructing meta-devices
It is supposed that the orthogonal circularly polarized incident waves can be described as Noting that the exterior rotation angle θ only produce opposite equally effect in two cross-polarized L-R and R-L channels, so the phase distribution produced by geometric phase modulation can be expressed as . Based on the corresponding relationship between Eq.
(S9) and (S10), the phase distributions for construction of meta-devices can be deduced by: Firstly, the phase distribution provided by propagation phase in co-and cross-polarized channels should satisfy the conditions in Eq. (S11a) and (S11c) simultaneously, where the initial two co-polarized and two cross-polarized channels are totally same. Then, in order to decouple the inherent consistent performance between co-polarized channels, the chirality-assisted phase pattern for co-polarized component should be designed to fit propagation phase distribution as Eq. (S11b), while it is noted that in Eq. (S11d) cross chiral  can produce decoupling effect in two cross-polarized fields. The last step is to apply exterior rotation to achieve independent wavefronts in two cross-polarized channels, where PB phase distribution is set as geo , each meta-atom at specific location can be designed and optimized to fit synthesized phase modulation, and desired meta-devices can be constructed.

Supplementary Note 4: Establishment of meta-atom library for measurface construction
In order to select the meta-atom to impose the preset phase distributions in all four CP conversion channels simultaneously, the library of meta-atoms is established with independent output phase profiles, as shown in Supplementary Figure 3. Based on the meta-atom library, the specific structure of meta-atom can be immediately obtained when the required four output phase profiles are fixed. It can be observed that with the specific interior angle ϑ, the output co-and cross-polarized phase profiles are functions of length px and width py of patch. When the interior angle changes, the phases in the two co-polarized channels exhibit distinct tendencies, while the phase response in the two cross-polarized channels are still kept similar, indicating that interior rotation can just decouple the inherent coherence between the two co-polarized outputs under LHCP and RHCP incidences.
Supplementary Figure 3. Meta-atom library with parameter sweeping of length px, width py and interior rotation angle ϑ. a Simulated co-polarized output phases and c cross-polarized output phases with LHCP incidence. b Simulated co-polarized output phases and d cross-polarized output phases with RHCP incidence. The solid sphere presents the different geometric sizes of patch layer and interior rotation angle, while the color shows the corresponding phase result.

Supplementary Note 5: Discussion on transmission amplitude of meta-atoms
In this part, the amplitude response of meta-atoms is discussed. Here, we apply four parameter degrees of freedom to manipulate all phase patterns of four transmission coefficients independently and simultaneously. The dimension sizes of patch layer (px and py) can originally set the phase profile of diagonal and off-diagonal elements, achieving independent φLL (= φRR) and φLR (= φRL). Then the interior angle ϑ can produce chiral responses in structure, which can decouple the phase response between φLL and φRR. After that, the exterior angle θ is applied to impose equal and opposite phase interruption in two cross-polarized components, resulting in the distinct response of φLR and φRL. These four degrees of freedom for decoupling the inherent consistency between four phase responses are the primary conditions within the whole phase modulation scheme.
It is noted that when there is no interior or exterior rotation (ϑ = θ = 0), the amplitude ratio between the diagonal and off-diagonal components can be determined by the phase difference between the linearly polarized phase responses along the fast-and slow-axes of birefringent meta-atom as illustrated in Eq.
(S4), which is guaranteed by px and py. Therefore, if the amplitudes are modulated by this linear-phase-difference factor, px and py would be fixed and limited by this condition. This means these two parameter degrees of freedom to define original phase pattern of diagonal and off-diagonal components would be invalid. Considering the primary condition for our proposed synthesis of three kinds of phase resources, phase modulation is more important than amplitude manipulation. Thus, the amplitude is indeed not considered and partly sacrificed during the optimization process to guarantee the phase requirements.
As mentioned in the main text, there are 25 meta-atoms collected to provide the required phase gradients in all four channels along x-direction. The calculated and simulated phase profiles of all meta-atoms are exhibited in Fig. 2b in the main text, while the corresponding simulated output amplitudes of CP conversion channels L-L. L-R, R-L, and R-R are illustrated in Supplementary Figure   4. It can be further observed that the amplitude responses of meta-atoms are not homogeneous, which is scarified to the phase limitations as explained above. We have to say that the non-uniform amplitudes would have some influence in energy distribution, such as the intensity mappings shown in Fig. 5a and 5b, where the doughnut-shape energy rings of vortex beams exhibit some heterogeneity. However, these unequal amplitude responses would not affect the phase distribution，meaning that the proposed phase-modulation scheme would not be limited by the sacrificed amplitude responses.
For future works, an extra degree of freedom needs to be explored to modulate the amplitude profiles so as to achieve independent manipulation of phase and energy in the four CP transmission channels.
Supplementary Figure 4. Simulated amplitudes of collected 25 meta-atoms for construction of proposed meta-deflector.