Abstract
1. INTRODUCTION
Freely customizing the properties and functions of available materials is the persistent pursuit of scientists, scholars, and engineers. Metasurfaces, which are planar arrays of subwavelength metallic or dielectric structures (meta-atoms), provide a robust way to customize properties because they allow an unprecedented degree of freedom (DOF) to manipulate electromagnetic (EM) waves [1–4]. By changing the geometrical parameters or orientation of meta-atoms, the amplitude [5], phase [6,7], and polarization [8] of the EM waves can be tailored to implement functional devices such as planar focusing lenses [9], planar retroreflectors [10,11], and holographic imagers [12,13]. Among them, metasurfaces have played very important roles in phase control and different kinds of phases can be obtained using metasurfaces.
In general, the phases imparted by metasurfaces mainly include four types: a resonance phase, a propagation phase, a detour phase, and a geometric phase. The resonance phase results from the localized electron oscillation within meta-atoms and is typically a Lorentzian type [14]. The propagation phase is an accumulated phase during electromagnetic wave propagation within materials that depends on the material thickness and wavelength [6,15]. The detour phase only appears under oblique incidence and is strongly dispersive both in the frequency domain and in the spatial domain [16,17]. The three kinds of phases above all result from localized resonances within or between meta-atoms. Therefore, they are all narrowband, which restricts their application in the development of broadband or even wideband functional devices. To obtain wideband phase control, the geometric phase is usually introduced by rotating meta-atoms by sequences. This type of phase is also known as a Pancharatnam–Berry (P-B) geometric phase and is usually nondispersive in a quite wide band [18–21]. P-B geometric phases have brought great impetus to the development of wideband functional devices using metasurfaces because of their wideband properties. Nevertheless, there is still an underlying dilemma for geometric phases obtained by rotating meta-atoms. A rotated meta-atom always imparts geometric phases with the same amplitude and opposite signs for left- and right-handed circularly polarized (LCP and RCP) waves. A P-B geometric phase is usually spin-coupled [6,21], which prohibits independent controls on LCP and RCP waves. To overcome this limitation, a common method combs the spin-coupled geometric phase with the spin-decoupled phase. In this way, the magnitudes or even the signs of geometric phases imparted to LCP and RCP can be changed. For example, bifunctional wavefront-manipulating devices are achieved by combining geometric phases and propagation phases [6,15,22,23]. Furthermore, other combinations, such as the resonance phase and the P–B phase [24–29], and the Aharonov–Anandan (A–A) phase and P–B phase [30,31], are also proposed. However, this scheme depends on the introduction of a “third-party” phase, and do not break the spin symmetry of the geometric phases. The geometric phases imparted to LCP and RCP waves are in fact still spin-coupled.
The question remains: does it come naturally to obtain spin-decoupled geometric phases using metasurfaces without combinations? In this paper, we explore a spin-decoupled geometric phase of a metasurface and propose to obtain spin-decoupled geometric phases by engineering the surface current paths on meta-atoms. To this end, the geometric phase obtained by rotating meta-atoms is first analyzed. It is found that the generation of a geometric phase by rotating meta-atom lies in the rotation of the surface current path. By calculating the rotational Doppler effect, we found the relationship between the current rotation and the geometric phase. The rotation angle of the current path determines the amplitude of the geometric phase, and the relative relationship between the rotation direction and the spin direction determines the sign of the geometric phase. The induced surface current paths under LCP and RCP waves always start oppositely and flow along the metallic structure; once the meta-atom rotates, the surface currents will rotate in the same direction. Because of the difference in the spin directions, the geometric phases should have the same magnitude and opposite signs when the meta-atoms are rotated, as is shown in Fig. 1(a). Within this in mind, to obtain spin-decoupled geometric phases, the induced surface current under one spin should be rotated by an angle while that under the other spin is rotated by different angles. In this way, LCP and RCP waves can acquire different geometric phase changes from a metasurface and a spin-decoupled geometric phase can be achieved using this kind of metasurface. As a proof of concept, we used a split-ring resonator (SRR) structure as the meta-atom to demonstrate this method. By changing the arc of the SRR, the rotation of the surface current on the SRR structure can be enforced without rotating the SRR meta-atom. Due to the actual rotation of surface current path, a geometric phase is naturally generated. As is depicted in Fig. 1(b), by separately customizing the start points of the surface currents induced by LCP and RCP waves, completely spin-decoupled geometric phases can be imparted to LCP and RCP waves by the metasurface. Prototypes were designed, fabricated, and measured. Both the simulation and experiment results verify spin-decoupled geometric phases and a spin-decoupled geometric phase can be readily combined with other phases such as a rotation geometric phase. This work provides a robust method to obtain spin-decoupled geometric phases and will further add to the metasurface’s DOF to manipulate EM waves.
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Figure 1.Two geometric phase modulations and their surface currents. The blue arrow represents the surface current
2. RESULTS AND DISCUSSION
A. Theory Analysis and Design
A geometric phase plays an important role in the development of a wideband metasurface due to its wavelength independent and nondispersive properties. Under circularly polarized (CP) wave illumination, a phase with span of and near-unity amplitude can be simultaneously obtained via a metasurface. The conventional way to obtain a geometric phase is by rotating the meta-atoms; that is, , where is the rotating angle, and represents LCP and RCP waves, respectively.
We choose the simplest dipole structure (a short straight metallic strip), as shown in Fig. 2, to analyze the essential reason to generate geometric phases by rotation. Previous studies have shown that the rotational Doppler shift occurs when a light beam carrying angular momentum propagates through a spinning object along its rotation axis and the Coriolis effect is responsible for it. When a coordinate system is attached to a rotated meta-atom, a rotational system is created. Since the rotating Doppler effect will be generated in the rotation system, we derive the rotational Doppler shift and the resulting geometric phase by establishing Maxwell’s equations under a rotating coordinate system. The geometric phase is expressed as
Figure 2.Geometric phase by rotation. (a) Schematic diagram of the Poincaré sphere. (b) Schematic diagram of the evolutionary path for the circular polarization on the Poincaré sphere. IN represents the incident polarization state, and OUT represents the reflected polarization state. The evolutionary path for the circular polarization on the Poincaré sphere must pass a point at the equator (middle row). Each column represents an evolutionary path. The top row is the RCP state, the middle row is the linear polarization state with a different polarization angle, and the bottom row is the RCP state. Green and purple arrows indicate the electric and magnetic fields, and black arrows show the polarization vectors, respectively [32].
As shown in Fig. 3, under the illumination of CP waves, a surface current will be induced on the metallic structure of a meta-atom. Once the meta-atom rotates, the surface currents must rotate with it. Since the secondary radiation of the surface current actually determines the EM properties of reflected fields, the P-B phase generated by rotation is essentially caused by the rotation of the surface current. In other words, the spatial orientation is the intrinsic reason for the geometric phase generation. Intuitively, the surface current induced by incident waves with different spin directions (LCP or RCP) will start and flow in opposite directions. In this way, when the metallic strip is rotated by angle , the surface current induced by LCP and RCP waves will also be rotated by angle relative to the center of the meta-atom. The three rotation states (, 60°, and 120°) are selected and presented in Fig. 3(a), which shows that, as the meta-atom rotates, the rotation angles of the surface currents under RCP and LCP wave illumination are 0°, 60°, and 120°. However, since the RCP and LCP waves spin in opposite directions, , where the and are the generated geometric phase under RCP and LCP wave illumination, respectively. The right panel of Fig. 3(a) is the simulated reflected phase in the three rotation states. Note that and exhibit opposite gradient directions. This feature is also known as the spin-locking of P-B phases.
Figure 3.Geometric phases generated by rotating meta-atom, where
Moreover, we found that when the meta-atom is rotated, the surface current at each point on the structure is also rotated around the center, as illustrated in Fig. 3(a). Therefore, we can convert the straight strip structure into an SRR with radius , as shown in Fig. 3(b). For the SRR structure, the induced surface currents under LCP and RCP waves always start from the two ends of the opening gap of SRR, respectively, and they also flow in opposite directions. When the SRR meta-atom is rotated, the induced surface currents are also rotated accordingly. The corresponding reflection phase is also given in Fig. 3(b) and is found to be consistent with that of the dipole structure.
The simulations and analysis indicate that the geometric phase is caused by the rotation of surface current on the meta-atom. The metasurface core is a metallic structure pattern design, which determines the detailed surface current path. This is why metasurface can be customized for EM manipulation. If the surface currents can be rotated by engineering the path, the geometric phase can also be obtained without rotating the meta-atom. The current diagram in Fig. 3(b) shows that the surface currents are induced by RCP and LCP wave flows at the two endpoints of the SRR gap. Considering that the two current paths flow at different starting points, once one endpoint rotates along the arc, the corresponding surface current will be forced to rotate with it, while another surface current induced by an orthogonal spin wave remains unchanged. In this way, the geometric phase gradient can be obtained for the former spin state but not for the other spin state. With this idea in mind, we define and as the angle between, respectively, the upper and lower ends of the opening gap referenced to the axis. They determine the start points of surface currents induced by the incident LCP and RCP waves. The schematic diagram of the meta-atom and the definition of the geometric parameters are shown in Fig. 4(a). The reflection performance of the meta-atom was first simulated. Circularly polarized (CP) waves are incident along the direction, while the and directions are set as “unit cell” boundary conditions. For more information about the simulation method, see Appendix E. As shown in Fig. 4(b), high-efficiency polarization conversion can be achieved within 8.0–14.0 GHz. Such polarization conversion results from the reversed propagation direction upon reflection. The near-unity reflection magnitude of the SRR meta-atom guarantees the design of a high-efficiency phase-modulation metasurface.
Figure 4.Spin-decoupled geometric phase obtained by engineering the surface current path on meta-atoms. (a) Schematic diagram of the SRR meta-atom, where
Surface current diagrams under different (40°, 85°, 130°, 175°) are given in Fig. 4(c), where it can be found that, as changes, the surface current induced by RCP waves is forced to undergo rotation with respect to the SRR circle center. In contrast, the surface current induced by RCP waves does not rotate because remains constant. The rotation angle of surface current coincides with the rotation angle of the SRR end point. Therefore, , , where and are the rotation angles of the surface currents induced by the RCP and LCP wave illumination. In this way, a reflection geometric phase is obtained, where . The corresponding reflection amplitude is depicted in Appendix B. The reflection phase, which varies with , is also given in Fig. 4(c). As changes, the reflection phase for the RCP waves can cover the span of , which is completely consistent with the theoretical predictions. On the other hand, the reflection phase for the LCP waves remains unchanged. When is kept unchanged and changes, LCP waves will obtain a geometric phase gradient while the RCP waves will not obtain a gradient. (For more details, see Appendix C.) Therefore, by engineering the surface current path on the meta-atom, a spin-independent geometric phase can be obtained, which will facilitate independent controls on the LCP and RCP waves.
B. Spin-Decoupled Metasurface and Experimental Verification
The example of the SRR meta-atom in the section above shows that by engineering the surface current path different geometric phases can be imparted to LCP and RCP waves. When only changes, only the RCP waves can acquire geometric phase changes (-geometric phase, for short), while the LCP waves do not. Similarly, when only changes, only the LCP waves can acquire geometric phase changes (-geometric phase, for short), while the RCP waves do not. Such a spin-decoupled geometric phase can be readily combined with other geometric phases, such as a conventional spin-coupled P-B phase by rotating the meta-atoms. When a whole meta-atom is rotated by angle , the LCP and RCP waves will additionally obtain opposite-signed geometric phases (-geometric phase, for short).
Figure 5.Design and simulated results of the three prototypes of a phase-gradient metasurface based on geometric phases. Note that the diffraction efficiency at the center frequency 11.0 GHz is calculated. (a) Completely spin-decoupled metasurface based on the
Since the generation mechanisms of the spin-decoupled -geometric phase and the spin-coupled -geometric phase are independent, it is natural to combine them to enhance the DOF in the phase modulation of the EM waves. Under such a consideration, we also designed a metasurface based on the combination of a -geometric phase and an -geometric phase, which can realize different functions for LCP and RCP waves. According to theoretical analysis, the geometric phases obtained by LCP and RCP waves are, respectively,
Furthermore, linearly polarized (LP) waves can be regarded as the superposition of LCP and RCP waves. The decomposition is expressed by the Jones vector [27]:
Figure 6.Deflection performance of the metasurfaces under
3. CONCLUSION
In conclusion, we completely explored the spin-decoupled geometric phase of a metasurface by engineering the surface current patch of meta-atoms. First, the generation mechanism of the spin-coupled geometric phase obtained by rotating the meta-atoms is analyzed, and it is shown that geometric phase generation lies in the rotation of the induced surface current. It is straightforward to rotate the surface current patch by rotating the meta-atom as a whole. However, such geometric phases are always spin-coupled, since both the surface currents induced by LCP and RCP waves are rotated by the same angle. Therefore, by rotating the meta-atom, geometric phases with the same magnitude but opposite signs can be imparted to the LCP and RCP waves. In fact, rotation of the surface current does not necessarily have to rotate the meta-atom. By engineering the start points of the surface current paths under the LCP and RCP waves, the rotation of the surface current also can be conveniently achieved. More importantly, the start point of surface current under the LCP waves can be made different from that under the RCP waves. In this way, the geometric phases imparted to the LCP and RCP waves by the metasurface can be completely decoupled. Prototypes were designed, fabricated, and measured. Both the simulation and measurement results verify this design philosophy. We believe our work provides a robust method to obtain spin-independent geometric phases and further increases the DOF for a metasurface to manipulate EM waves.
APPENDIX A: THE DERIVATION OF ANGULAR DOPPLER SHIFT
First, we consider the differential form of Maxwell’s equations in an inertial frame [
We define a rotating frame with the angular velocity in the lab frame; velocity, . represents the transformation of the vector in the rotating frame. Transformation of the coordinate system does not result in transformation of the variables [
When a charge of quantity is moving with velocity , charge conservation reveals that the change in the charge and current in a rotating coordinate system is
The charge is thus observed in a rotating coordinate system at rest and is subjected to an electric field force:
Hence, the transformation of the electric field is
The force density on current in a neutral conductor in the lab frame is for this to equal the force density in the rotating frame. For a passive field, where there is no charge, we can get
Then, we consider time-varying vector , and the derivative of with respect to time can be expressed as
For the lab frame,
For an observer at rest in the rotating frame, ; thus
It can be obtained from Eqs. (
Considering that we generally analyze uniform rotation, .
In rectangular coordinates [
From Eq. (
With this analysis in mind, we now consider the form of Maxwell’s equations in rotational coordinates.
According to Eqs. (
The vector is replaced by and , then
Then,
Finally, if we assume that the space in question has no free charges and no free currents, we get the form of Maxwell’s equations in a rotating coordinate system:
The wave equation is transformed as [
Assuming that the incident plane wave is propagating along the direction, some vertical components are ignored to simplify the wave equation for analysis. The total electric field is the sum of the horizontal () and vertical components () of the electric field; i.e., , :
Then the Eq. (
Assuming that the incident plane wave is circularly polarized, the electric field is expressed as
The and are defined as and . Thus, Eq. (
In this case, the frequency shift of the waves is found in the rotating coordinate system, which is called the Doppler angular shift [
Under such conditions, when a beam of light is incident on a rotating object, the scattered light must have a certain phase difference with the incident light [
APPENDIX B: THE REFLECTION AMPLITUDE OF THE rRL AND rLR UNDER β1 CHANGING
As shown in Fig.
Figure 7.Reflection amplitude of the
APPENDIX C: THE MODULATION OF THE RCP VIA β2 CHANGING
As shown in Fig.
Figure 8.Geometric phase generation for the LCP wave. (a) Surface current paths under different
Figure 9.Reflection phase and amplitude of the 24 meta-atoms: (a)
APPENDIX D: THE PARAMETERS AND REFLECTION PERFORMANCE OF META-ATOMS DEPICTED IN FIG.?5
The reflection performance and parameters of the 24 meta-atoms are depicted in Fig. Parameters of 24 Meta-Atoms Depicted in Fig. Meta-Atom Number No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 40 167.5 115 62.5 190 137.5 0 30 60 90 120 150 85 212.5 160 107.5 55 182.5 0 30 60 90 120 150 130 77.5 205 152.5 100 47.5 0 30 60 90 120 150 175 122.5 70 197.5 145 92.5 0 30 60 90 120 150
APPENDIX E: THE SIMULATION METHOD
CST Microwave Studio commercial software is used to simulate the performance of the meta-atoms and metasurface. With the help of the CST field monitor, the surface current and far-field beam can be calculated and depicted in the manuscript. For the meta-atom simulation, the Frequency Domain Solver is used and the boundary conditions along the and directions are set as the unit cell while two Floquet ports are fixed along the direction.
For the metasurface simulation, the meta-atoms are arrayed to construct a metasurface using CST Microwave Studio software. The Time Domain Solver is applied and the boundary conditions along the , , and directions are set as the “Open Add Space.” The plane wave has the linear and circular polarization states applied as the excitation source to illuminate onto the metasurface.
References
[7] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, Z. Gaburro. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science, 334, 333-337(2011).
[34] B. A. Garetz. Angular Doppler effect. J. Opt. Soc. Am., 71, 609-611(1981).
[37] K. T. McDonald. Electrodynamics of Rotating Systems(2008).
[38] E. A. Bondarenko. Two systems of Maxwell’s equations and two corresponding systems of wave equations in a rotating dielectric medium. Electromagnetic Fields and Waves(2019).
[39] W. M. Irvine. Electrodynamics in a rotating system of reference. Physica, 30, 1160-1170(1964).
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