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

Aug 21, 2020

### Principle of phase manipulation of full CP channels

Here, we propose a general formalism for complete and separate phase manipulation of all four CP channels, where arbitrary and independent wavefronts can be achieved by altering the CP states of input and output ends, as schematically illustrated in Fig. 1. Taking generation of OAM modes as a demonstration example, four OAM mode with topological charge l = 0, 1, 2, and 3 are realized through different CP channels. The corresponding theoretical spatial phase distributions of the different output OAM modes and the schematic CP conversion process from input to output waves are exhibited in the insets of Fig. 1.

Different from previous works on birefringent metasurfaces41,45, the aim of this work is to perform distinct phase-modulation in four CP transmission channels, which can be described by the four element transmission coefficients in the Jones matrix (T_{{mathrm{cir}}} = left[ {begin{array}{*{20}{c}} {t_{{mathrm{LL}}}^{}} & {t_{{mathrm{LR}}}^{}} \ {t_{{mathrm{RL}}}^{}} & {t_{{mathrm{RR}}}^{}} end{array}} right]). The equivalent metasurface system is supposed to be passive, lossless, matched, and reciprocal. Therefore, the four CP transmission coefficients (the first/second subscript represents the input/output CP state, L/R denotes the LHCP/RHCP state) with linear base can be described as follows:

$$t_{{mathrm{LL}}} = frac{1}{2}left[ {( {t_{{mathrm{xx}}} + t_{{mathrm{yy}}}} ) + {mathrm{i}} cdot ( {t_{{mathrm{xy}}} – t_{{mathrm{yx}}}} )} right],$$

(1a)

$$t_{{mathrm{LR}}} = frac{1}{2}left[ {( {t_{{mathrm{xx}}} – t_{{mathrm{yy}}}} ) – {mathrm{i}} cdot ( {t_{{mathrm{xy}}} + t_{{mathrm{yx}}}} )} right] cdot {mathrm{e}}^{{mathrm{i}} cdot 2theta },$$

(1b)

$$t_{{mathrm{RL}}} = frac{1}{2}left[ {( {t_{{mathrm{xx}}} – t_{{mathrm{yy}}}} ) + {mathrm{i}} cdot ( {t_{{mathrm{xy}}} + t_{{mathrm{yx}}}} )} right] cdot {mathrm{e}}^{ – {mathrm{i}} cdot 2theta },$$

(1c)

$$t_{{mathrm{RR}}} = frac{1}{2}left[ {( {t_{{mathrm{xx}}} + t_{{mathrm{yy}}}} ) – {mathrm{i}} cdot ( {t_{{mathrm{xy}}} – t_{{mathrm{yx}}}} )} right],$$

(1d)

where (t_{{mathrm{xx}}} = left| {t_{{mathrm{xx}}}} right| cdot {mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{xx}}}}) and (t_{{mathrm{yy}}} = | {t_{{mathrm{yy}}}} | cdot {mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{yy}}}}) are the diagonal linear transmission coefficients, and (t_{{mathrm{xy}}} = | {t_{{mathrm{xy}}}} | cdot {mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{xy}}}}) and (t_{{mathrm{yx}}} = | {t_{{mathrm{yx}}}} | cdot {mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{yx}}}}) are the off-diagonal linear transmission coefficients. θ is the exterior rotation angle introduced by rotation matrix (M(theta ) = left[ {begin{array}{*{20}{c}} {cos theta } & {sin theta } \ { – sin theta } & {cos theta } end{array}} right]). Here, tLL and tRR are defined as co-polarized transmission channels, which maintain the polarization state of input waves. tLR and tRL represent cross-polarized channels, which flip the output fields into opposite CP state. The first components with totally same expression in tLL and tRR (or tLR and tRL) can be labeled as (t_{{mathrm{propa}}}^{{mathrm{co}}} = | {t_{{mathrm{propa}}}^{{mathrm{co}}}} |{mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{propa}}}^{{mathrm{co}}}} = frac{1}{2}(t_{{mathrm{xx}}} + t_{{mathrm{yy}}}))(( {t_{{mathrm{propa}}}^{{mathrm{cross}}} = | {t_{{mathrm{propa}}}^{{mathrm{cross}}}} |{mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{propa}}}^{{mathrm{cross}}}} = frac{1}{2}( {t_{{mathrm{xx}}} – t_{{mathrm{yy}}}} )} )). The phase pattern of the two transmission components (varphi _{{mathrm{propa}}}^{{mathrm{co}}}) and (varphi _{{mathrm{propa}}}^{{mathrm{cross}}}) are uniquely dependent to propagation phase modulation, which would produce initial influence in both co-polarized and cross-polarized fields regardless of the incident CP state. Meanwhile, the second components carrying opposite symbols in tLL and tRR (or tLR and tRL) can be extracted as (t_{{mathrm{chiral}}}^{{mathrm{co}}} = | {t_{{mathrm{chiral}}}^{{mathrm{co}}}} |{mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{chiral}}}^{{mathrm{co}}}} = frac{1}{2} cdot {mathrm{i}} cdot (t_{{mathrm{xy}}} – t_{{mathrm{yx}}}))(( {t_{{mathrm{chiral}}}^{{mathrm{cross}}} = | {t_{{mathrm{chiral}}}^{{mathrm{cross}}}} |{mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{chiral}}}^{{mathrm{cross}}}} = frac{1}{2} cdot {mathrm{i}} cdot ( {t_{{mathrm{xy}}} + t_{{mathrm{yx}}}} )} )). The phase pattern of these two components (varphi _{{mathrm{chiral}}}^{{mathrm{co}}}) and (varphi _{{mathrm{chiral}}}^{{mathrm{cross}}}) are determined by chirality-assisted phase, indicating that the two components would be an additional degree of freedom to decouple inherent consistency between co-polarized channels. Moreover, the PB phase pattern generated by rotating matrix φgeo = 2θ, which is sensitive to the incidence polarization state, influence only the two cross-polarized fields. Here, the plus or minus (±) sign represents the conjugate responses to LHCP or RHCP input states. According to Eq. (1a–d), it can be concluded that the full phase-modulation scheme is based on three phases that constitute three degrees of freedom to decouple the inherent consistencies between CP transmission coefficients: (i) propagation phase modulation is applied to define the initial phase profiles of two diagonal and two off-diagonal transmission elements tLL (=tRR) and tLR (=tRL) when there is no chirality-assisted and geometric phase responses, (ii) the chirality-assisted phase modulation is introduced to decouple the consistency between tLL and tRR while keeping tLR and tRL unchanged, and (iii) the geometric phase modulation can further realize distinct profiles of tLR and tRL, meanwhile producing no effect on tLL and tRR.

When the proposed metasurface is illuminated by LHCP incident wave (| {overrightarrow {mathrm{L}} } rangle = left[ {begin{array}{*{20}{c}} {mathrm{1}} \ {mathrm{i}} end{array}} right]) and RHCP incident wave (| {overrightarrow {mathrm{R}} } rangle {mathrm{ = }}left[ {begin{array}{*{20}{c}} {mathrm{1}} \ {{mathrm{ – i}}} end{array}} right]) respectively, the output electric fields can be expressed as:

$$overrightarrow E _{{mathrm{out}}}^{{mathrm{L,in}}} = T cdot left| {mathrm{L}} rightrangle \ = left( {t_{{mathrm{propa}}}^{{mathrm{co}}} + t_{{mathrm{chiral}}}^{{mathrm{co}}}} right) cdot left| {mathrm{L}} rightrangle + left( {t_{{mathrm{propa}}}^{{mathrm{cross}}} – t_{{mathrm{chiral}}}^{{mathrm{cross}}}} right) cdot {mathrm{e}}^{{mathrm{i}} cdot varphi _{{mathrm{geo}}}} cdot left| {mathrm{R}} rightrangle \ = overrightarrow E _{{mathrm{LL}}} + overrightarrow E _{{mathrm{LR}}}$$

(2a)

$$overrightarrow E _{{mathrm{out}}}^{{mathrm{R,in}}} = T cdot left| {mathrm{R}} rightrangle \ = left( {t_{{mathrm{propa}}}^{{mathrm{co}}} – t_{{mathrm{chiral}}}^{{mathrm{co}}}} right) cdot left| {mathrm{R}} rightrangle + left( {t_{{mathrm{propa}}}^{{mathrm{cross}}} + t_{{mathrm{chiral}}}^{{mathrm{cross}}}} right) cdot {mathrm{e}}^{ – {mathrm{i}} cdot varphi _{{mathrm{geo}}}} cdot left| {mathrm{L}} rightrangle \ = overrightarrow E _{{mathrm{RR}}} + overrightarrow E _{{mathrm{RL}}}$$

(2b)

It can be observed from Eq. (2a, b) that under the orthogonal CP illuminations, the metasurface can produce four different output fields, including two co-polarized output components (overrightarrow E _{{mathrm{LL}}}) and (overrightarrow E _{{mathrm{RR}}}), and two cross-polarized components (overrightarrow E _{{mathrm{LR}}}) and (overrightarrow E _{{mathrm{RL}}}). Based on the analyses above, four CP transmission coefficients can be independently phase-modulated, where the inherent consistencies between four output phase patterns φLL, φLR, φRL, and φRR are independently decoupled by propagation phase, chirality-assisted phase and geometric phase, respectively.

### Meta-atom design and verification

Here a passive, reciprocal transmission-type meta-atom is applied to implement all these three phase modulation schemes, namely chirality-assisted phase, propagation phase, and PB phase. The proposed meta-atom is composed of five metallic layers and four dielectric substrates for realizing a 4th-order LC band-pass filter, whose geometric structure of meta-atom is introduced and schematically shown in Fig. 2a and the detailed operating mechanism is explained in Supplementary Note 1, Supplementary Figs. 1 and  2. In our designed meta-atom, propagation phase is provided by tailoring the width px and length py of the rectangular patch layers as shown in Fig. 2b. Figure 2c exhibits the phase profiles of all four outputs against px, where the phase response of the two co-polarized output channels show the same tendency φLL = φRR, as well as the two cross-polarized output phase delays φLR = φRL, while the co-polarized and cross-polarized phase responses exhibit differently φLL ≠ φLR (φRR ≠ φRL) (the co-polarized and cross-polarized phase profile tendencies against both px and py are shown in Supplementary Fig. 3). That is to say, propagation phase can achieve independent manipulation of co-polarized and cross-polarized phase delays but does not show any selectivity on the spin state of incidence. In order to conquer this limit, chirality-assisted phase is introduced by the relative rotation of three similar patch layers of meta-atom with different angles, regarded as an interior rotation. Here orientation of middle layer is fixed with no further rotation, while the upper and lower layers are rotated with opposite angles ϑ respectively, as shown in Fig. 2d. A representative meta-atom selected with specific dimensions and different interior angle ϑ is simulated and presented in Fig. 2e. It can be seen that the two co-polarized phase profiles exhibit distinct tendency φLL ≠ φRR, while the two cross-polarized phase responses keep same tendency with the interior rotation φLR = φRL, which is due to the symmetrical rotation angle in upper and lower patch layers (derived and explained in Supplementary Note 2 and Supplementary Note 3). It is verified that the chirality-assisted phase can decouple the intrinsic coherence between co-polarized output under orthogonal CP states, while makes no disparate effect in two cross-polarized channels. As for the geometric phase, it is performed through rotation of the whole meta-atom structure with angle θ named exterior rotation as shown in Fig. 2f. Figure 2g presents the four output phase profiles of the representative meta-atom with specific dimensions and interior angle but various exterior angles, where two cross-polarized phases exhibit different tendency φLR ≠ φRL and two co-polarized phases are the same φLL = φRR.

Based on all above theoretical and simulated results, it can be confirmed that all four phase responses of the CP channels can be separately tuned with independent geometrical parameters of the proposed meta-atom. For meta-devices pursuing complete manipulation of the four CP conversion channels to achieve multiple functionalities simultaneously, the meta-atom library to endow the required phase gradients of all CP conversion channels, is established with sweeping of length px, width py and interior angle ϑ presented in Supplementary Note 4. Once the required phase profiles in different channels are confirmed, meta-atoms can be selected from the library and further optimized to construct the desired multifunctional metasurfaces. It is noted that, in this scheme we applied four parameter degrees of freedom (px, py, ϑ, and θ) to independently manipulate four phase patterns of transmission channels (φLL, φLR, φRL, and φRR). However, within this phase-modulation process, the amplitude responses are not considered and the efficiency is partly sacrificed due to the particularity of the proposed meta-atom structure. This limitation is detailed in Supplementary Note 5 and Supplementary Fig. 4. The exploration of an extra degree of freedom (e.g., material losses, active components, and even time) for independent amplitude modulation is therefore desired in future works.

### Simulation and measurement results on two metasurfaces

Based on the above specific explanation of general principle and construction of the meta-atom library, here we briefly clarify the construction of a single metasurface to achieve a distinct wavefront from each of the four CP channels. It is supposed that the desired spatial phase distributions can be described by (F_{{mathrm{LL}}}(x,y)), (F_{{mathrm{LR}}}(x,y)), (F_{{mathrm{RL}}}(x,y)), and (F_{{mathrm{RR}}}(x,y)) respectively. To implement the synthesized three phase schemes, chirality-assisted phase, propagation phase and PB phase components should be established by desired spatial phase distributions simultaneously (detailed in Supplementary Note 3). Following the phase modulation scheme, here we construct two metasurfaces with distinct functionalities to demonstrate the proposed phase responses for independent manipulation of full CP conversion channels.

As a first device concept, a meta-deflector able to separate all the co-polarized and cross-polarized output beams to four independent directions, is designed and the functional schematic is shown in Fig. 3a. The phase distribution of the proposed meta-deflector is based on the generalized Snell’s Law and is described as:

$${mathrm{{Phi}}}_i^{{mathrm{refr}}}(x,y) = {Delta}{Phi}_i cdot x = frac{{2{uppi}}}{{lambda _0}}(sin varsigma _i^{{mathrm{out}}} – sin varsigma ^{{mathrm{in}}}) cdot x,,\ {mathrm{with}},i = 1,,2,,3,,{mathrm{and}},4,$$

(3)

where the normal direction is set along +z-axis and ({Delta}{Phi}_i) represents the phase gradient along x-axis. ςout and ςin are the angles of output and incident waves relative to +z-axis, and we consider the normal incident case where (varsigma ^{{mathrm{in}}} = 0^circ). In this section, the output tilting angles are preset as (varsigma _{mathrm{1}}^{{mathrm{out}}} = – 35^circ) in L–L channel, (varsigma _{mathrm{2}}^{{mathrm{out}}} = 0^circ) in L–R channel, (varsigma _{mathrm{3}}^{{mathrm{out}}} = 58^circ) in R–L channel and (varsigma _{mathrm{4}}^{{mathrm{out}}} = – 16^circ) in R–R channel. The corresponding required phase gradients are ({Delta}{Phi}_1 = frac{{uppi }}{{mathrm{3}}}) for FLL(x,y), ({Delta}{Phi}_2 = 0) for FLR(x,y), ({Delta}{Phi}_3 = -frac{{ {uppi}}}{{mathrm{2}}}) for FRL(x,y), and ({Delta}{Phi}_4 = frac{{uppi }}{{mathrm{6}}}) for FRR(x,y). As for the construction of metasurface, the desired phase gradients for different channels along x-axis are discretized and implemented by 25 meta-atoms. The theoretical phase profiles and simulated phase value of all these meta-atoms are exhibited in Fig. 3b, indicating the feasibility of metasurface construction. Figure 3c displays the simulation and measurement results of the meta-deflector for four independent refracted wavefronts, whose far-field patterns at the center frequency of 10 GHz and the results clearly describe the peak direction of the output field intensities. It can be seen that under LHCP wave illumination, the co-polarized output wavefront tilts to −35° (in L–L channel), while the cross-polarized component indeed does not show any refraction effect (in L–R channel). Upon flipping the incidence into RHCP, the LHCP and RHCP output components are reconstructed with deflected wavefronts of 58° and −15° as verified by simulation and measurement (in R–L and R–R channel), respectively. It is worth-noting that in this part, the cross-polarized output wavefront under LHCP incident wave is preset with refraction angle of 0°, indicating that the energy in this channel is manipulated with converted polarization and kept unchanged wavefront state with the input. Another meta-deflector design is presented in the Supplementary Note 8 and Supplementary Fig. 6 to verify that all four output wavefronts can be imposed with different and arbitrary refraction angles. For further demonstration, the meta-deflector is fabricated and experimentally measured in microwave region as detailed in “Methods” section, A photograph of the sample is shown in Fig. 3d and measured results are displayed in Fig. 3e. The far-field pattern measurements have further been performed in frequency band from 9 to 11 GHz, which are function against frequency and detection angles. It can be seen that the deflecting angle slightly changes within the whole bandwidth, which can be considered as the frequency dispersion in Eq. (3). However, at the center frequency of 10 GHz, all the simulated and measured wavefronts generated in channels L–L, L–R, R–L, and R–R present the desired refraction directions, verifying the availability of proposed mechanism for simultaneous and independent manipulation of full spin conversion channels.

Additionally, we further examine the evolution of diffraction order against the variation of incident polarization state. Any arbitrary polarized wave κ can be considered as the superposition of two orthogonal CP waves with different proportionality coefficient, as described as:

$$kappa = eta _{mathrm{L}} cdot left| {mathrm{L}} rightrangle + eta _{mathrm{R}} cdot left| {mathrm{R}} rightrangle,$$

(4)

where ηL and ηR denote the coefficient of LHCP and RHCP components. Here, we select five different polarization states as the incident wave, and the polarization variation path can be described as the black line on the Poincaré sphere in Fig. 4. From the north pole (point A) representing RHCP to the south pole (point E) for LHCP, continuous evolution is experienced through right-handed elliptical polarization (RHEP, point B), linear polarization (x-LP, point C), and left-handed elliptical polarization (LHEP, point D), where the ratios between ηL and ηR of these five polarized incidence A, B, C, D, and E are 0, 0.3, 1, 3, and ∞.

The simulated and measured output far-field intensities for five input polarization states are exhibited in Fig. 4. When the input polarization is RHCP, the output far-field intensity contains two main lobes, co-polarized component R–R with −16° refraction and cross-polarized component R–L with 58° refraction, as shown in Fig. 4a. With the polarization of incidence changing to state B, it can be seen from Fig. 4b that the output wavefront includes four lobes located at approximately, −35°, 0°, 58°, and −16° directions, which are the individual contributions of the four output channels L–L, L–R, R–L, and R–R. The proportion between the sum of energy distributed in both L–L and L–R channels to that in R–L and R–R channels is approximately 1:3, corresponding to the ratio between LHCP and RHCP component at the input end with RHEP state. Figure 4c displays the far-field intensity under LP wave illumination, where the far-field lobes in L–L and L–R channels exhibit nearly equal amplitude to that in R–L and R–R channels. When the input state is switched to LHEP, the proportion of output sum energy in both L–L and L–R channels to that in R–L, R–R channels is changed to 3:1 and displayed in Fig. 4d, as reversely symmetric situation to that in RHEP results. Finally, under LHCP wave illumination, only two main lobes with −35° and 0° refraction are left as in Fig. 4e. According to the above evolution process, it can be confirmed that all four CP channels can be fully utilized, and the ratio between different diffraction orders can be considered as function of the incident polarization.

For further demonstration of complete manipulation of the four CP conversion channels, a spin-to-orbital angular momentum meta-convertor is proposed and designed, where the orbital angular momentum (OAM) with different topological charge is achieved in each CP channel. The spatial helical phase distribution of optical spiral phase plate for OAM generation is described as:

$${Phi}_i^{{mathrm{OAM}}}(x,y) = l_{mathrm{i}} cdot arctan (y/x),{mathrm{with}},i = 1,,2,,3,,{mathrm{and}},4,$$

(5)

where (x, y) is the required coordinate of meta-atom and li = i − 1 is the topological charge of OAM mode. ({Phi}_i^{{mathrm{OAM}}}(x,y)) with i = 1, 2, 3, and 4 are the corresponding phase distributions for (F_{{mathrm{LL}}}(x,y)), (F_{{mathrm{LR}}}(x,y)), (F_{{mathrm{RL}}}(x,y)), and (F_{{mathrm{RR}}}(x,y)), respectively.

Figure 5a displays the simulated energy distributions in xoy plane at z = 150 mm (corresponding to 5λ0 for the center frequency of 10 GHz) of transmitted output vortex beams carrying OAM modes with l = 0, 1, 2, and 3 through the CP channels L–L, L–R, R–L, and R–R orderly, where all the typical doughnut-shape energy rings are exhibited and compared. It can be obviously seen that in L–L channel, the OAM mode is 0, whose energy intensity is as a plane wave with no hollow distribution in the center. Meanwhile with the topological charge increasing from l = 1–3 in L–R, R–L, and R–R channels, respectively, the radius of energy rings is gradually enlarged, which experimentally verifies the divergent properties of typical vortex beam. Figure 5b presents the corresponding measured energy distributions of vortex beams generated from the CP conversion channels, exhibiting four vortex beams with increasing energy ring radius, which is in good agreement with the simulations. Meanwhile, it can be seen that there exist some nonuniform and discrepancies in simulated and measured energy distributions, which are detailed discussed in Supplementary Note 6. The simulated and measured phase distributions of vortex beams carrying the corresponding four OAM modes are shown in Fig. 5c, d. It can be obviously observed that the helical phase pattern of OAM modes are totally distinct, verifying that the functionalities in all four CP channels can be independently modulated and the helical patterns are in accordance with the topological charge l · 2π. The measured phase patterns agree well with the simulated results, which further indicates the feasibility of independent multiple OAM modes generation based on full CP conversion channels.

Here, it should be noted that the phase manipulation to achieve distinct responses in all four transmission CP channels is a general method, which is not limited by operating wavelength. To extend the method to shorter wavelengths, efforts in high precision lithography would be required. Moreover, metallic lossy plasmonic resonators to be used in the replacement of current metallic patches, would have to be replaced by chiral all-dielectric nanoantennas.