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Abstract
The geometric phase in metasurfaces follows a symmetry restriction of chirality, which dictates that the phases of two orthogonal circularly polarized waves are identical but have opposite signs. In this study, we propose a general mechanism to disrupt this symmetric restriction on the chirality of orthogonal circular polarizations by introducing mirror-symmetry-breaking meta-atoms. This mechanism introduces a new degree of freedom in spin-decoupled phase modulation without necessitating the rotation of the meta-atom. To demonstrate the feasibility of this concept, we design what we believe is a novel meta-atom with a QR-code structure and successfully showcase circular-polarization multiplexing metasurface holography. Our investigation offers what we believe to be a novel understanding of the chirality in geometric phase within the realm of nanophotonics. Moreover, it paves the way for the development of what we believe will be novel design methodologies for electromagnetic structures, enabling applications in arbitrary wavefront engineering.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurfaces have emerged as a versatile and compact platform for wavefront engineering in the last decade [1,2]. These ultrathin electromagnetic (EM) functional layers enable overall wavefront engineering through the linear superposition of EM waves that are modulated by each meta-atom, acting as the unit cell of the metasurface. A meta-atom's phase modulation mechanisms typically involve resonant phase, dynamic phase, detour phase, and geometric phase. The resonant phase is the rapid phase shift near the resonant wavelength in meta-atoms. It is caused by the resonance and contributes a phase shift of π. [3]. The dynamic phase arises from accumulated phases of an EM wave propagating in a meta-atom, influenced by factors such as its refractive index, geometry, and other meta-atom parameters [4,5]. The detour phase arises from the displacement of adjacent elements, enabling phase modulation in the propagation direction different from the incident direction [6]. On the other hand, the geometric phase, also known as the Pancharatnam-Berry phase, arises from the spin-orbit coupling of photons in a meta-atom and is dependent on the rotation angle of the anisotropic EM structures such as elliptical and rectangular structures [7–9]. For instance, when a circularly polarized EM wave passes through such structures with a rotation angle θ, the transmitted cross-polarized component exhibits the geometric phase Φ. Conventionally, Φ = ±2θ, where the sign ± is determined by the chirality of the EM wave, such as left- and right-circular polarizations (LCP and RCP, respectively). The geometric phase is symmetrical for both orthogonal circular polarizations. Hence, achieving spin-decoupled EM wavefront engineering solely through the geometric phase proves to be challenging.
In recent years, researchers have made efforts to disrupt the symmetry restriction on the chirality of geometric phase in various ways. For instance, nonlinear effects have been introduced to rewrite the geometric phase of transmitted harmonic waves as ±(n ± 1)θ, where n represents the order of harmonic generation [10–12]. However, this approach is not capable of arbitrarily manipulating the geometric phases of orthogonal circular polarizations. In the context of linear processes, Bai et al. proposed a method that combines the Aharonov-Anandan phase dependent on the meta-atom structures and the PB phase dependent on the rotation angles of the meta-atoms to overcome the limitation of symmetry in the geometric phase [13]. This proposed method is applicable in engineering metasurfaces for EM wave manipulation [14,15]. Jin et al. realized focusing both spins of light independently by utilizing both the geometric phase and dynamic phase [16]. Another approach, suggested by Chen et al., involves the use of dielectric planar chiral meta-atoms analyzed through an electric dipole model. This approach enables local phase manipulation and controlled spin decoupling of orthogonal circularly polarized waves [17]. Moreover, Jin et al. proposed a solution for metallic metasurfaces using v-shaped chiral meta-atoms, which facilitate phase decoupling for the two eigen spin states [18]. Recently, Xiong et al. proposed to break the polarization-multiplexing limit by engineering the noise of the solution of Jones matrix elements, enabling up to 11 independent holograms through one single metasurface [19]. These studies have explored various physical mechanisms for creating asymmetric geometric phases and have demonstrated spin-decoupled phase modulation. However, while these investigations have examined specific decoupling mechanisms on a case-by-case basis, there remains a need for a comprehensive understanding of this field at a philosophical level. Specifically, further exploration is required to investigate the relationship between meta-atom topology and the breaking of symmetry in the chirality of phase retardations for orthogonal circular polarizations. Such investigations are of great significance in deepening our comprehension of the symmetry of geometric phase in meta-atom structures, as well as the corresponding properties of electromagnetic wavefront engineering.
In this study, the restriction of symmetry in the geometric phase is analytically disclosed for the first time, to the best of our knowledge, as originating from the mirror-symmetry of EM structures. To elaborate, from a geometric and topological perspective, the absence of mirror symmetry in the meta-atom topology results in the breaking of symmetry in the chirality of two orthogonal circularly polarized waves. It is important to note that different geometric structures can entail distinct polarization-decoupling mechanisms. Furthermore, we demonstrated a new degree of freedom (DOF) in modulating geometric phases through non-mirror-symmetric topologies, facilitating spin-decoupled phase modulation without the necessity of meta-atom rotation. To substantiate these findings, we devise and examine non-mirror-symmetric meta-atoms incorporating QR-code structures to verify the violation of the symmetry restriction on chirality of orthogonal circular polarizations. As a proof of concept, we design a multiplexed hologram utilizing circularly polarized waves employing a metasurface composed of QR-code meta-atoms. Through numerical simulations, we validate our proposed theory. These findings contribute to a novel comprehension of the geometric phase and light-matter interactions in the realm of nanophotonics, paving the way for more advanced applications in metasurface- and metamaterial-based technologies.
2. Theory of geometric phase for non-mirror-symmetric meta-atoms
First, we derive a general form of the geometric phase for a meta-atom with an arbitrary topology, such as a non-mirror-symmetric structure. When an EM wave is incident on a meta-atom, the Jones matrix can be expressed in the circular polarization basis [20,21] by defining the basis vectors $\hat{L} = \frac{{\sqrt 2 }}{2}({\hat{x} - i\hat{y}} )$ for LCP, and $\hat{R} = \frac{{\sqrt 2 }}{2}({\hat{x} + i\hat{y}} )$ for RCP, respectively. The Jones matrix S for circular-polarization base vectors is (see details in Appendix A)
Further, if the meta-atom rotates by an angle θ, then the Jones matrix J is expressed as $J(\theta )= R({ - \theta } )JR(\theta )$, where
We substitute the formula of J(θ) into Eq. (1), and obtain the most general form of the geometric phase for an arbitrary meta-atom using circular-polarization base vectors as follows:
Compared to the linear Jones matrix J(θ), the matrix S(θ) demonstrates several significant properties in an intuitive manner. Firstly, the diagonal elements S11 and S22 remain unchanged regardless of the rotation of the meta-atom. Secondly, the anti-diagonal elements S21 and S12, which represent the transmitted cross-polarizations, naturally break symmetry. In other words, the cross-polarization transmissions of incident circularly polarized EM waves are inherently asymmetric, challenging the conventional viewpoint. Traditional meta-atoms with topologies like rectangular, elliptical, and other structures, which have been studied extensively in previous research, constrain the chirality of geometric phases due to their mirror-symmetric topologies, resulting in J12 = J21 = 0. Consequently, the asymmetric elements S21(θ) and S12(θ) degenerate into well-known symmetric geometric phase forms:
It is important to mention that even though the geometric phases described in Eq. (4) exhibit a symmetric form, it is still feasible to decouple the phase retardations of two orthogonal circular polarizations by combining both the dynamic and geometric phases. In order to enhance the understanding of this notion, we reformulated Eq. (4) in a more intuitive manner, presented below:
From Eqs. (3)–(5), it can be concluded that the conventional symmetry restriction on the chirality of orthogonal circular polarizations is believed to stem from the mirror symmetry exhibited by meta-atoms. Consequently, breaking the underlying mirror symmetry of the meta-atoms inevitably results in the breakdown of chiral restrictions.
Next, we rewrite S21(θ) and S12(θ) in Eq. (3) in a more intuitive form. If we simplify S21 and S12 as ${A_{21}}{e^{i{\varphi _{21}}}}$ and ${A_{12}}{e^{i{\varphi _{12}}}}$, respectively, with A21 (A12) and φ21 (φ12) denoting the amplitude and argument angle of S21(S12), then we can express S21(θ) and S12(θ) as:
3. Schematic of the non-mirror-symmetric meta-atom with QR-code structure
In this study, we present a novel dielectric meta-atom featuring a QR-code structure, aiming to eliminate any geometric symmetry, as shown in Fig. 1. This QR-code topology, previously explored in our investigation of perfect metamaterial absorbers, exhibits a significant number of DOF, thereby offering extensive capabilities for wavefront engineering [26]. The meta-atom employed in this research comprises a silica substrate and arrays of square silicon pillars arranged in the form of a QR code, with a periodicity (P) of 800 nm. Each QR code encompasses a grid of 5 × 5 pixels, where each pixel is randomly assigned as either a square silicon pillar (length: 100 nm; height: 1400 nm) or air. Due to the completely random nature of the generated QR-code topology, mirror-symmetry is disrupted easily and thoroughly. Fabrication of the proposed QR-code structure can be carried out on a fused silica substrate by employing state-of-the-art techniques such as Si deposition, electron beam lithography, and inductively coupled plasma etching. Notably, the utilization of the clear-oxidize-remove-etch sequence in recent advancements enables the achievement of high-aspect-ratio nanopillars with smooth, straight sidewalls [27].
Fig. 1. Schematic of the non-mirror-symmetric meta-atom with QR-code structure: (a) 3D view and (b) top view. The parameters are set as P = 800 nm, L = W = 100 nm, and H = 1400 nm.
We conducted full-wave 3D finite-difference time-domain (FDTD) simulations to design and investigate the structures of QR-code meta-atoms. The refractive index of the silica substrate was set at 1.45, and the dielectric constant values of silicon were derived by fitting Palik's experimental data to the Drude-Lorentzian model [28,29]. The period of the QR code meta-atoms was set to 800 nm, while the light source was set to 1550 nm. This ensured that the meta-atom array remained non-diffractive in both air and the silica substrate [30]. For the simulation of meta-atoms, period boundaries were applied in the horizontal direction, and perfectly matched layers (PMLs) were used in the longitudinal direction. By randomly assigning values to each pixel of the QR-code structure, we obtained a diverse range of non-mirror-symmetric meta-atoms and numerically calculated their parameters, which is discussed in the next section.
4. Breaking the symmetry restriction of chirality using non-mirror-symmetric meta-atoms
We conducted 3D FDTD simulations to design and investigate the QR-code meta-atom structures. We designed and numerically evaluated 13,000 meta-atom structures consisting of a random arrangement of 5 × 5 QR-code nanopillars. All the QR-code meta-atoms exhibit decoupled geometric phases for LCP and RCP. To ensure efficient polarization-multiplexing applications, we selected structures with transmission coefficients exceeding 0.7. The corresponding S21 and S12 of these selected structures are depicted in Fig. 2(a). Notably, the phase retardations of the LCP and RCP cover a full range of 0–2π and break the symmetry restriction, which aligns with the predictions of Eqs. (3) and (6). For the sake of simplicity, yet without loss of generality, the meta-atoms in Fig. 2 are not rotated, indicating a rotation angle θ of zero. These results indicate that the spin-decoupled phase modulations, arising from non-mirror-symmetric structures, introduce novel DOF for engineering EM wavefronts. Furthermore, we successfully engineered the geometric phases independently for orthogonal circular polarizations within the entire range of 0 to 2π.
Fig. 2. Phases of cross polarizations for different meta-atom structures. (a) Phases of cross polarizations for the QR-code structures shown in Fig. 1. (b) Phases of cross polarizations for different mirror-symmetric structures. (c) Top views of mirror-symmetric meta-atoms investigated in (b). Each plot includes the geometrical parameters for the corresponding meta-atom structure, with units in nm.
We investigated a variety of mirror-symmetric meta-atom structures as a control group, and their S21 and S12 are plotted in Fig. 2(b). All mirror-symmetric meta-atoms have the same period (800 nm) and height (1400 nm) as the non-mirror-symmetric ones, but different topologies in the top view, which are depicted with detailed geometric parameters in Fig. 2(c). Regardless of the meta-atom topology, be it rectangle, ellipse, trapezoid, cross, T-shap, H-shape, or hollow rectangle, all geometric phases are positioned along the y = x line. This indicates that the geometric phases of the two orthogonal circular polarizations possess equivalent values. This behavior aligns with the anticipated symmetry restriction outlined in Eq. (4) and arises as a consequence of the mirror-symmetric structures.
Figure 3 illustrates the comparison of electric field intensity distributions between meta-atoms with mirror-symmetry and those without mirror-symmetry. In Figs. 3(c)-(f), the optical modes for one circular polarization in conventional mirror-symmetric meta-atoms exhibit mirror symmetry with the modes of the orthogonal circular polarization. This observation aligns with the theoretical analysis presented in Eqs. (4) and (5), which elucidate the existence of a symmetry constraint on the chirality observed in the phase retardations of orthogonal circular polarizations. Conversely, for the non-mirror-symmetric meta-atom topology depicted in Figs. 3(a) and (b), the optical modes under illumination with orthogonal circular polarizations manifest distinct behaviors without any symmetry. This characteristic corresponds well with the theoretical predictions in Eqs. (3) and (6), resulting in the breaking of symmetry in the chirality of orthogonal circular polarizations.
Fig. 3. Electric field intensity distributions in the top view plane passing through the center of non-mirror-symmetric meta-atoms in (a) and (b), and mirror-symmetric meta-atoms in (c)-(f). (a), (c), and (e) show the right-circularly polarized components under left-circularly polarized incidence, while (b), (d), and (f) depict the left-circularly polarized components under right-circularly polarized incidence. The dashed lines in each subplot depict the top view of the meta-atom topologies in each case.
From Figs. 2 and 3, our analytical predictions of the geometric phases can be verified. Specifically, the decoupling of the geometric phases can be achieved when the mirror-symmetry is broken. Additionally, the spin-decoupled phase retardations, which is independent of the rotation angle of the meta-atom, introduces a new DOF in wavefront engineering.
As a proof of concept, we designed a circular-polarization multiplexing metasurface hologram based on our proposed QR-code meta-atoms. The design of the circular-polarization-decoupled metasurface hologram is depicted in Fig. 4 (Flowchart). Firstly, the phase-only computer-generated holograms (CGHs) were calculated for incident light beams that were left- and right-circular polarized. To accomplish this, we employed the Gerchberg-Saxton (GS) algorithm [31]. We generated two different images for the two orthogonal polarizations and obtained two output holographic phase diagrams after the GS iterations.
Next, we applied an eight-level phase quantization to convert the two phase diagrams into two phase matrices. In this procedure, we selected 8 × 8 meta-atom structures from Fig. 2(a) to establish the eight-level phase library. This library was then searched to identify the meta-atom structures satisfying the phase requirements for both the LCP and RCP holograms. To achieve this, we performed a one-by-one element search on the phase matrices.
Lastly, we simulated the desired metasurface hologram using 64 types of QR-code structures. Due to limited computational resources, we conducted a proof of concept simulation that involved a metasurface with 101 × 101 periods. The simulation successfully validated our theory. For this proof of concept, the period size was fixed at 800 nm, allowing the metasurface hologram to cover an area of 80.8 × 80.8 µm2.
The designed metasurface hologram was simulated using 3D FDTD, and the reconstructed images for LCP and RCP are presented in Fig. 5. Figures 5(a) and (e) show the 101 × 101-pixel dog and cat images as the original images, which are illuminated by left- and right-circularly polarized light, respectively. Thus, the images are reconstructed using the phase retardations that incorporate the cross-polarization components RCP and LCP. The phase-only CGHs of the two orthogonal polarizations, simulated using the GS algorithm, are show in Figs. 5(b) and (f). As the control group, the images of the theoretical reconstructions are presented in Figs. 5(c) and (g). The reconstructed images of the simulated metasurface hologram, which was illuminated by different circularly polarized EM waves, are plotted in Figs. 5(d) and (h). It can be observed that the reconstructed images of the metasurface hologram exhibit good agreement with the original and theoretically reconstructed images, thereby confirming the feasibility of decoupling circularly polarized light via wavefront engineering. In addition to visual observation, the quality of the reproduced image can also be characterized by quantitative criteria. In the field of holography, the correlation coefficient is often used to measure the fidelity of holographic reconstruction image [32]. A coefficient value of 1 indicates a fidelity of 100%. For the LCP and RCP illuminations, the correlation coefficients between the reconstructed images and the original images were found to be 0.91 and 0.88, respectively. These results signify a strong correlation between the holographic reconstructions and the input images, thereby validating the high quality of the metasurface holography. This result demonstrates that non-symmetric meta-atom structures can indeed break the symmetry restriction of chirality for spin-decoupled phase modulations, providing new insights into metasurfaces and metamaterials and promoting their application in various fields. Furthermore, the scheme of breaking the symmetry restriction of chirality based on non-mirror-symmetric meta-atom topologies is independent from other methods such as detour phase engineering [6], noise engineering [19], etc. Thus, by combining the design of non-mirror-symmetric meta-atom with other schemes, it is potential to realize multiple polarization-multiplexing channels independently in future work.
Fig. 5. Circular-polarization-decoupled metasurface holography. (a) and (e) are the original images for LCP and RCP illuminations, respectively. (b) and (f) correspond to the phase distributions of the calculated holograms. (c) and (g) are the images of the theoretical reconstructions. (d) and (h) correspond to the reconstructed images of the simulated metasurface hologram.
5. Conclusion
In conclusion, we have theoretically established a general formula to describe the geometric-phase property of a meta-atom using the Jones matrix and circular-polarization base vectors. The analytical results indicate that the non-mirror-symmetry of meta-atom leads to the breaking of the symmetry constraint on the chirality of orthogonal circular polarizations. To illustrate this, we have designed a novel QR-code meta-atom that breaks the topological mirror symmetry and enables spin-decoupled phase modulation. Additionally, we have demonstrated that the non-mirror-symmetry topology of meta-atoms provides a new DOF in wavefront engineering. This results in spin-decoupled phase modulation without the need to rotate the meta-atom. Further, we have designed a circularly polarized multiplexed metasurface hologram using our proposed QR-code meta-atoms. This design showcases the potential of symmetry-broken phase modulations in various applications in metasurfaces and metamaterials. We believe that this study will expand the understanding of the geometric phase in a fundamental way and will facilitate the development of design methodologies for EM structures, which can be used for arbitrary wavefront engineering.
Appendix A. Derivation of Jones matrix S
When a left-circularly polarized light beam is incident on an arbitrary meta-atom, the transmitted light is
Similarly, when a right-circularly polarized light beam is incident, the output light field is expressed as:
We can rewrite the above equations using linear-polarization base vectors. For a left-circularly polarized incident light beam, the transmitted electric field is given by
Similarly, for a right-circularly polarized incident light beam, we obtain
The physical processes represented by the aforementioned equations remain identical irrespective of using a circular- or linear-polarization base vector. Thus, we solve Eqs. (7)–10) simultaneously and establish the analytical relationship between J and S as follows:
When the meta-atom rotates by an angle θ, the corresponding Jones matrix J can be written as follows
We substitute Eq. (12) into Eq. (11), and obtain the Jones matrix S for a rotated meta-atom
Appendix B. Mirror-symmetry of meta-atoms
For an x-polarized incident light beam, the output light wave is given by
If a mirror operation is performed on the meta-atom along the symmetric axis (for example, the x-axis), then the Jones matrix $J^{\prime}$ becomes $\left( {\begin{array}{{cc}} {J_{11}^{\prime}}&{J_{12}^{\prime}}\\ {J_{21}^{\prime}}&{J_{22}^{\prime}} \end{array}} \right)$, and the output becomes
According to the property of mirror operation, the y-polarization component effectively undergoes a phase retardation of ${\textrm{e}^{i\pi }}$, and thus, we obtain ${J_{11}}\hat{x} = J_{11}^{\prime}\hat{x}$, ${J_{21}}\hat{y} = J_{21}^{\prime}\hat{y} \cdot {e^{i\pi }}$, which implies $J_{11}^\mathrm{^{\prime}} = {J_{11}}$, $J_{21}^\mathrm{^{\prime}} ={-} {J_{21}}$.
Similarly, in the case of the y-polarization incident light beam, we can obtain $J_{12}^{\prime} ={-} {J_{12}}$, $J_{22}^{\prime} = {J_{22}}$. Thus, the relationship between J and $J^{\prime}$ can be expressed as
For a mirror-symmetric meta-atom, mirror operation on its symmetric axis does not change its geometry. Thus, we obtain following equation:
Further, if the mirror-symmetric meta-atom rotates by an angle θ, then the corresponding Jones matrix S(θ) in Eq. (13) is simplified to
Funding
National Natural Science Foundation of China (12374307, 12234009); National Key Research and Development Program of China (2023YFA1406903, 2022YFA1404800).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
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