Synergetic full-parametric Aharonov–Anandan and Pancharatnam–Berry phase for arbitrary polarization and wavefront control

2.1 Concept and fundamentals

2.1.1 AA phase mechanism

The realization of arbitrary polarization conversion through Aharonov–Anandan (AA) phase demands precise control over both amplitude and phase responses of each Jones matrix element. While previous chiral metamaterials have achieved spin-dependent phase manipulation, our framework advances this paradigm by analyzing the relationships within the Jones matrix components of chiral meta-atom and achieving elegant amplitude-phase control. Firstly, we analyze a reflective meta-atom’s EM properties through the linear Jones matrix J lin = r x x e x x r x y e x y r y x e y x r y y e y y in Cartesian base. Here, |r _xy | (φ _xy ) and |r _yy | (φ _yy ) represent reflection amplitude (phase) responses of x- and y-polarized components under excitation of y-polarized wave (the same nomenclature to other parameters). Then, the J_lin can be transformed into circular base as J_cir (J_cir = Λ⁻¹J_linΛ) through transformation factor Λ = 1 2 1 1 i − i , and simplified according to symmetry [56], lossless and unitarity [46], [57] features (see more details in Supporting Information Section S1)

(1) J cir = r l l e φ l l r l r e φ l r r r l e φ r l r r r e φ r r = r x x cos Δ ⋅ e i Φ 1 − r x x cos Δ 2 e i arctan ∓ r y x r x x sin Δ e i Φ 1 − r x x cos Δ 2 e i arctan ± r y x r x x sin Δ e i Φ r x x cos Δ ⋅ e i Φ

where Δ = φ x x − φ y y / 2 , and Φ = φ x x + φ y y / 2 . This analysis yields a universal conclusion: the sum of phases in co-polarized components equals that of two cross-polarized scattering components ( φ l l = φ r r = φ r l + φ l r 2 ), which is applicable to systems with symmetry, unitarity, and losslessness. Here, we investigate the scenario characterized by phase φ _rl = φ_AA and φ _lr = 0, where φ_AA represents AA phase in σ r l induced by changing arc angle α of the arm. Finally, Jones matrix J_AAL with AA phase of meta-atom is presented as below:

(2) J AAL = r l l e i φ AA 2 1 − r r r 2 1 − r l l 2 e i φ AA r r r e i φ AA 2

A basic chiral meta-atom prototype was designed and numerically characterized to validate above theory, as shown in inset (i) of Figure 1b. Figure 2 and Table S1 presents the reflected amplitude and phase response of the proposed meta-atom under normal LCP wave illumination, demonstrating |r _rl | > 0.75 across 9–14 GHz while satisfying the lossless condition (|r _ll |² + |r _rl |² ≈ 1). Similar behavior is observed under RCP wave excitation (Figure 2d), where |r _lr | = |r _rl | and |r _rr |² + |r _lr |² ≈ 1. A larger phase cover of 270° with quasi-nondispersive properties is achieved in σ r l ( φ r l = φ AA ≈ 2 ∫ d Δ α ) by tuning α with φ _lr fixed to zero (Figure 2b, c, and e), indicating that the meta-atom exhibits broadband spin-dependent phase shift across 9–14 GHz. Although the AA phase is intrinsically wavelength-independent, the meta-atoms required for parameter control may introduce undesired material dispersion and structural resonances which would limit the operational bandwidth. Moreover the theoretical calculations indicate that phase variation in σ l l is half of those in σ r l (Figure 2f), and this conclusion also applies to phase of σ r r . These results are in good agreement with those calculated by theoretical derivation in Eq. (2), validating our proposed theory of AA phase. Additionally, the proposed methodology can be universally extended to achieve simultaneous phase shifts in both σ l r and σ r l through spin-dependent geometric paths engineering of meta-atom, as theoretically derived in Eqs. (S5) and (S6) (Supporting Information).

Figure 2:

EM characterization of proposed chiral meta-atom based AA phase mechanism. Numerically calculated reflection magnitude and phase response at 9–14 GHz for (a–b) σ r l and (c) σ l r under LCP(RCP) wave illumination. (d) The amplitude and (e) phase responses across co-polarized ( σ l l / σ r r ) and cross-polarized ( σ l r / σ r l ) component at 10 GHz. (f) The phase shifts and its ratio between co- and cross-polarized components (Δφ _ll and Δφ _rl ) by tuning α from 40° to 140° at 10° intervals under LCP wave excitation at 10 GHz.

2.2 Arbitrary LP wave control using spin-decoupled phase manipulation

To achieve arbitrary LP-to-LP conversion and wavefront control, we implement spin-decoupled phase manipulation via meta-atoms (Figure 3a) featuring simultaneous modulation of arc angle α for AA phase control and orientation angle θ for PB phase engineering. As indicated from Eq. (S7), this equation can be further simplified as J PB  +  AAL = 0 e i 2 θ e i φ AA − 2 θ 0 with φ _rl = φ_AA − 2θ and φ _lr = 2θ when |r _lr | = |r _rl | = 1, which evidently demonstrates that phases of φ _rl and φ _lr are spin decoupled. The polarization angle of arbitrary LP-to-LP conversion is governed by ψ = (φ _rl – φ _lr )/2 = φ_AA/2−2θ, while the associated phase retardation (Δφ = φ_AA/2) dictates the polarization-dependent wavefront shaping capability. Figure 3b–e present simulated amplitude (|r _rl | and |r _lr |) and phase (φ _rl and φ _lr ) responses of two types of LP conversion meta-atoms libraries (meta-atoms A and B) under LCP or RCP wave illumination, satisfying the spin-decoupled condition of |r _lr | = |r _rl | ≈ 1 and full 2π phase cover with quasi-nondispersive performance across 9–14 GHz. The polarization angle of reflected wave approximates 45° and −45° under excitations of y-LP wave (Figure 3d and g), respectively. To verify the proposed method, a focusing polarization conversion metasurface was designed using meta-atoms established in Figure 3. As demonstrated in Supplementary Figure S6, high-quality focusing is achieved across 9–14 GHz (43.5 % fractional bandwidth), confirming the non-dispersive behavior in metasurface with synergetic AA and PB phase mechanisms. Furthermore, multi-focus planar lens integrating two types of distinct meta-atoms A and B with polarization and phase control capabilities was designed via complex-phase addition theorem [58] and diagonally interleaved method. Orthogonal output polarized states between meta-atoms A and B ensures negligible polarized crosstalk. To enhance resolution, an interleaved coding was employed where each pair of meta-atoms (A and B) with periodicity p _r ( p r = 2 p ) forms a minimal functional unit, see Figure 4a. The corresponding EM response (amplitude, phase, polarization) of meta-atoms and phase distributions are detailed in Figure S11 (Supporting Information). The phase φ ^mn for each meta-atom of multi-focus planar lens meta-device can be calculated through complex-phase superposition formula ( φ m n = arg ∑ i = 1 4 e − j φ i m n ), where φ i m n denotes the phase of the mn-th meta-atom required to generate i-th focusing beam. Ultimately, y-polarized waves are converted into ±45° LP states after impinging the meta-device and focused on distinct spatial positions (Figure 4a).

Figure 3:

Structure and EM response of spin-decoupled meta-atoms with synergetic AA and PB phase mechanisms. (a) Structural evolution of meta-atoms implementing from geometric phase to synergetic AA and PB phases, where AA phase is tuned via arc angle α while PB phase is controlled through orientation angle θ, respectively. Numerically calculated amplitude (|r _rl | and |r _lr |) and phase response (φ _rl and φ _lr ) of cross-circular polarization components for meta-atoms A (b–c) and B (e–f) under LCP and RCP wave illumination, where meta-atoms A and B represent structures transforming y-polarized wave to 45° and −45°-LP waves, respectively. Polarization angle of meta-atoms for y-polarized wave to (d) 45° and (g) −45° LP waves, where ψ = (φ _rl – φ _lr )/2.

Figure 4:

Experimental characterization of the spin-decoupled planar multifocal lens for arbitrary LP states. (a) Meta-device array layout, where dashed line represents real periodicity p _r of meta-atoms A and B. (b) Photograph of the fabricated sample of planar multifocal lens. (c) Near-field experimental setup. (d–e) Simulated and measured normalized 2D electric field intensity |E_45°| and |E_−45°| at 11.1 GHz in xoy plane (z = 204 mm), with cross-sectional profiles along y at x = −53, 0, 0, and 50 mm.

For experimental demonstration, a porotype of planar multifocal lens was fabricated with sample shown in Figure 4b and characterized using the near-field experimental setup illustrated in Figures 4c and S12a, where LP horn antenna and sample were turned around 45° clockwise. Two horizontally and vertically oriented waveguide probes were employed to capture reflected static EM signals from −45° and 45° LP waves, respectively. Upon interaction with meta-atom A, the y-polarized incident wave was transformed into a −45° LP wave, generating four localized energy foci at coordinates (−53, −49.5 mm), (−53, 49.5 mm), (0, −56 mm), and (0, 56 mm) within xoy plane (z = 204 mm). These measured foci characterized by 0.6 power beam width of field pattern normalized to the maximized intensity exhibited a size of 8.3, 19.3, 19.4 and 2.8 mm, respectively. Similarly, meta-atom B achieved conversion of y-polarized waves to 45° LP state with energy foci at (0, −55.5 mm), (0, 55.5 mm), (50, −56.5 mm), and (50, 56.5 mm) in the same plane. The corresponding focal sizes are 16.6 mm, 24.9 mm, 11.1 mm, and 19.3 mm. Simulated and experimental focal positions at 11.1 GHz align closely with theoretical predictions, as demonstrated in Figure 4d and e. Furthermore, two-dimensional (2D) electric field distributions in yoz planes at x = −53 mm, 0 mm, and 50 mm (Figures S9 and S10) clearly illustrate the designed multi-focus behavior with high spatial resolution and cross-polarization suppression across 10–12 GHz. The discrepancy between the simulated and measured multiple foci observed in Figure 4e may have resulted from differences in the selection of the focal plane (Figures S9 and S10). Relatively narrower bandwidths between meta-atoms in Figure 3 and array originate primarily from phase errors induced by near-field coupling and resonant frequency shift of meta-atoms as the periodicity increases from p to p _r (Figures S11 and S12). Additionally, laterally distributed multifoci inherently induce chromatic dispersion under broadband operation, resulting in undesired focal elongation along x axis. The measured focusing efficiency, defined as the ratio between the powers carried by the focal spot of 45° (−45°) LP and the incident beam, was calculated as 91 % (89 %). Large signal intensity is received based on simultaneous matching of polarization states and spatial position, confirming the meta-device’s capability for arbitrary LP-to-LP conversion and high-precision wavefront control.

2.3 Arbitrary polarization and wavefront control using amplitude-phase control

To enable arbitrary polarization detection and communication, the polarization state of incident EM wave needs to be converted into arbitrary states of polarization. Here, we demonstrated the conversion of CP states to arbitrary states of polarization alongside wavefront control via the amplitude-phase manipulation. As shown in Eq. (S4), the co-polarized phase response follows φ _rr = φ_AA/2, while the cross-polarized phase shifts are governed by θ₁ + θ₂ under RCP wave illumination. This configuration enables precise polarization engineering through the derived polarization angle ψ = φ_AA/4 − (θ₁ + θ₂)/2 and phase retardation Δφ = φ_AA/4 + (θ₁ + θ₂)/2. The ellipticity manipulation of the anisotropic diatomic meta-atom is dictated by the orientation angle difference Δθ = θ₂ − θ₁, establishing a mapping between geometric parameter space and full-Stokes polarization control. Furthermore, we validate the applicability of this strategy through the design and characterization of functional meta-device. According to the relationship between arbitrary polarization parameters and components of CP wave ( χ = 1 2 si n − 1 A r 2 − A l 2 A r 2 + A l 2 and Ψ = φ r − φ l 2 ) [44], the elliptical angle χ and polarization angle ψ of EM wave can be collaboratively manipulating the amplitude ratio A _l /A _r and phase difference of two orthogonal CP components, where A _r and A _l denote reflection magnitude for σ l r and σ r r , respectively. Although phase cover of σ r l can be extend to 2π through altering α, phase of σ l l is only half of that in σ r l according to Eq. (S4) in Supporting Information. Theoretically, the additional π phase jump in σ l l or σ r r can be continued by further merging the AA phase in σ l r component, as illustrated in Eq. (S6) (Supporting Information). Here, 1-bit coding diatomic meta-atoms presented in inset (iii) of Figure 1b were adopted to design meta-device that convert RCP wave into left-handed elliptically polarized wave (ψ = 90°, χ = 13.5°), while simultaneously satisfying both the phase cover and the amplitude ratio A _l /A _r at 9.96 GHz (Figure 5a). The relatively narrower bandwidth observed in Figure 5a compared to the meta-atoms in Figure 3 results from multi-resonance behavior, and spectral shifts in both co- and cross-polarized components induced by parameter adjustments. The detailed geometric parameters of diatomic meta-atoms are α = 46.6°, θ₁ = −52.1°, and θ₂ = −16.8° for φ = 0, and α = 137.8°, θ₁ = 17.1°, and θ₂ = 59.8° for φ = π. Simulations reveal that both types of meta-atoms exhibit minor amplitude and phase variations under 30° oblique incidence (Figure S13, more details in Supporting Information Section S5), where edge-positioned meta-atoms of the array exerting negligible influence on the scattered wave’s polarization state.

Figure 5:

Experimental characterization of diatomic meta-device with amplitude-phase control for arbitrary polarization and wavefront manipulation. (a) Simulated amplitude (|r _lr | and |r _rr |) and phase (|φ _lr | and |φ _rr |) responses of the 1-bit meta-atoms under RCP wave illumination. (b) Photograph of the fabricated sample. (c) Far-field experimental setup. (d) Amplitude ratio and (e) phase difference between LCP and RCP components. (f–h) Simulated and experimental 1D far-field scattering patterns at f = 10 GHz in xoz plane. Here (f), (g), and (h) correspond to that of total, LCP, and RCP components of scattered wave.

For proof-of-principle verification, a 15 × 15 beam-steering diatomic meta-device was designed with linear phase distribution shown in Figure S14. The porotype was fabricated with sample shown in Figure 5b and measured using the far-field experimental setup shown in Figures 5c and S10b. For quantitative and intuitive characterization, the linear amplitude ratio and phase difference of σ r r and σ l r under RCP wave excitation are firstly calculated, see Figure 5d and e. Numerical and experimental elliptical angles of main beam coincide well and are observed as 13.6° and 13.4° at 10 GHz for total component, respectively. Since background scattering in nonideal measurement setup inherently perturbs the radiation pattern of meta-device II, the sidelobes below −12.3 dB exhibit high sensitivity to scattering. However, this nonideal scattering pose has little effect to the main beam (exhibits robustness to such perturbations) due to concentrated energy of main beam. This sharp contrast accounts for large and minor deviations between simulated and measured LCP/RCP component ratio observed in sidelobe at θ = 30° and main beam. Moreover, the simulated polarization angle of 94.2° Ψ = φ r l − φ l l 2 is very close to that (90°) of target polarization state, indicating that incident CP wave is successfully converted into left-handed elliptically polarized wave. Figure 5f–h compare simulated and measured far-field scattering patterns whose deflection angles of main beam are observed as 20.5° and 23°, respectively which are close to the predicted one of 20°. The LCP and RCP components exhibit different scattered intensity, and they combine in space to form an elliptical polarization state. The measured half-power beamwidth of main beam was 8.1°, and the sidelobe level was −12.3 dB. The anomalous deflection efficiency, defined as ratio between the power of the main beam and totally reflected power, is calculated as 83.1 %. To further validate the control relationship for co-polarized component derived in Eq. (S8), we designed an additional polarization-preserving metasurface simultaneously maintained incident polarization and imparted desired wavefront phases, validating the amplitude-phase control strategy (Figure S15). To sum up, the above results demonstrate that the diatomic meta-device can effectively convert RCP wave into left-handed elliptical CP wave and thus validate the feasibility of arbitrary polarization control based on our proposed AA phase.