High-efficiency terahertz spin-decoupled meta-coupler for spoof surface plasmon excitation and beam steering

Spoof surface plasmon (SSP) meta-couplers that efficiently integrate other diversified functionalities into a single ultrathin device are highly desirable in the modern microwave and terahertz fields. However, the diversified functionalities, to the best of our knowledge, have not been applied to circular polarization meta-couplers because of the spin coupling between the orthogonal incident waves. In this paper, we propose and numerically demonstrate a terahertz spin-decoupled bifunctional meta-coupler for SSP excitation and beam steering. The designed meta-coupler is composed of a coupling metasurface and a propagating metasurface. The former aims at realizing anomalous reflection or converting the incident waves into SSP under the illumination of the left or right circular polarization waves, respectively, and the latter are used to guide out the excited SSP. The respective converting efficiency can reach 82% and 70% at 0.3THz for the right and left circular polarization incident waves. Besides, by appropriately adjusting the reflection phase distribution, many other diversified functionalities can also be integrated into the meta-coupler. Our study may open up new routes for polarization-related SSP couplers, detectors, and other practical terahertz devices.

Recently, polarization-controlled bifunctional SSP and surface plasmon (SP) meta-couplers are proposed and experimentally demonstrated in the microwave and optical regimes [29][30][31]. These works open up new routes for multifreedom-controlled SSP/SP excitation. However, most of them are designed with symmetrical or similar functionalities, not reaching the goal of realizing independent multiple functionalities. SSP/SP couplers that integrate multiple diversified functionalities into a single ultrathin device are highly desirable in the modern microwave, terahertz, and optical fields. In 2017, by using the unit cells with different reflection phase, Cai et.al. design high-performance reflective bifunctional metasurfaces with independent beam steering and SSP excitation for orthogonal linear-polarization incidence in the microwave regime [32]. Similar functionalities are also realized for transmissive metasurfaces with independent anomalous reflection and beam focusing. In addition, at visible wavelength, Ding et.al. use gap-plasmon metasurfaces to implement polarizationcontrolled unidirectional SP excitation and beam steering at normal incidence [33]. Similarly, polarization-switchable multi-functional metasurface for focusing and SP excitation are also designed by Ling et.al. [34]. All these works realize the independent SSP/SP excitation and beam steering under Figure 1. Schematic view of the bifunctional meta-coupler that can realize CP-controlled unidirectional SSP excitation and anomalous reflection. The meta-coupler consists of a coupling metasurface and a propagating metasurface. orthogonal linear polarization electromagnetic wave (EMW) illumination. However, meta-couplers that integrate multiple diversified functionalities, to the best of our knowledge, have not been demonstrated for circular polarization (CP) incidence up to now because of spin coupling between the orthogonal incident waves [35,36].
In this work, we design and demonstrate a high-efficiency CP-controlled bifunctional meta-coupler for independent SSP excitation and anomalous reflection. The bifunctional meta-coupler is composed of a coupling metasurface and a propagating metasurface, as illustrated by Fig. 1. By combining the resonant phase with the Pancharatnam-Berry phase, efficient decoupling can be realized between orthogonal CP incident waves [37][38][39]. In this way, arbitrary independent phase gradients can be achieved theoretically. By elaborately adjusting the phase distribution of the coupling metasurface, the incident waves can be converted into SSP with longitudinal wave vector kx = 1.2 k0 or anomalously reflected with reflection angle θr = 13.9º under the illumination of the left circular polarization (LCP) or the right circular polarization (RCP) beam, respectively. Finite element method (FEM) full-wave simulations show that the respective converting efficiency of the RCP and LCP at 0.3 THz can reach up to 82% and 70% and their corresponding half-power bandwidth are 0.0285 and 0.01 THz. Besides, we further demonstrate that the meta-coupler can maintain high efficiencies (>48%) with the incident Gaussian beam waistwidth ranging from 0.7λ to 1.7λ, which verifies the robustness of the designed bifunctional meta-coupler. In addition to the anomalous reflection, many other diversified functionalities can also be integrated into the SSP metacoupler in this way.

II. THEORY ANALYSIS
We now describe the physical mechanism of spin-decoupling between orthogonal CPs. First of all, we analyze the electromagnetic properties of the unit cells of the metasurface. Modeling the reflective metasurface as a oneport device, its reflection properties can be characterized by To simplify the complexity of the further analysis, we design the unit cells with mirror symmetry structures. This can efficiently eliminate the cross coupling between orthogonal linear-polarization incident waves, i.e. rxy = ryx = 0 [36]. In this case, the Jones matrice (0) R can be written as where the figure "0" in bracket represents the rotation angle of the unit cells. When we rotate the unit cells at an arbitrary angle θ, a necessary rotation transformation will be made on the original matrice. Utilizing the rotation operator where  ( ) θ R represents the Jones matrice of the unit cell with a rotation angle θ under the circular polarized mode base, and † ( ) θ Q represents the transposed conjugation of the rotation operator ( ) θ Q . Assuming that the reflection coefficient xx r and yy r meet the relation rxx=-ryy, then the Jones matrice For the CP incident waves with opposite handness, i.e. . Compared with the incident waves, we can find that, in addition to the polarization transformation, two opposite geometrical angles are encoded as phase shifts (Pancharatnam-Berry phase) into the reflective CP waves. The opposite phase shifts for the orthogonal CPs lead to symmetrical functionalities, which is also called photonics spin hall effect (PHSE) [35,36]. To break the symmetry and realize the independent phase control for orthogonal CP incidence, another coefficient rxx is considered as an additional variable to control the reflected waves. Based on the previous assumption rxx=-ryy, we further consider that rxx=ryy = e jφ , which means that each unit cell has a specified resonant phase with uniform electric field amplitude. To meet this condition, the losses of the metal and dielectric spacer are small enough and the working frequency is beyond the resonant absorption band. In this work, the amplitude of the reflection coefficient of the unit cells in use is all higher than 0.99, as illustrated by Fig. 3(b). In this way, another resonant phase φ is introduced as a new degree of freedom. Therefore, the Jones matrice can be expressed as Then the reflected waves under the LCP and RCP incidence can be expressed as ( 2 ) 0 respectively. By elaborately adjusting the resonant phase φ and the rotation angle θ, arbitrary independent reflection phase can be obtained for orthogonal CP wave incidence.

A. DESIGN OF THE COUPLING METASURFACE
To realize CP-controlled SSP excitation and beam steering, 2D arrays of unit cells with different resonant phase and rotation angles are required to constitute the coupling metasurface. In ths work, f = 0.3 THz is chosen as the central working frequency and the period of the coupling unit cells is set as p = 233µm. Besides, we select Δφ = 60º and Δθ = 20º so the total reflection phase difference between the adjacent coupling unit cells for the LCP and RCP incidence is 100º and 20º, respectively. Of course, the selections of Δφ and Δθ are arbitrary as long as they meet the conditions that |Δφ-2Δθ| > k0p > |Δφ+2Δθ| or |Δφ+2Δθ| > k0p > |Δφ-2Δθ|, where k0 represents the wave vector in free space. For the RCP incidence, the longitudinal wave vector of reflected waves is kr = (Δφ -2Δθ) / p = 0.24 k0. Therefore, the anomalous angle θr = arcsin (kr / k0) = 13.9º under normal incidence. For the LCP incidence, the longitudinal wave vector of reflected waves is kr = (Δφ+2Δθ) / p = 1.2 k0, which means that the surface waves are excited on the coupling metasurface. By designing a propagating metasurface with the same Eigen wave vector and electric field modes, the excited surface waves can be guided out with high efficiency. We now design the practical unit cells that meet all the theoretical requirements mentioned above. The unit cells are composed of metal patches and flat metal mirror separated by a 50 μm-thick dielectric spacer Taconic RF-43 with the The permittivity of copper is characterized by a Drude model ε(ω) = 1 − ωp 2 ∕ (ω 2 + iωγ), where ω is the angular frequency, ωp = 1.123 × 10 16 Hz is the plasma frequency and γ = 1.379 ×10 13 Hz is the collision frequency [40,41]. We adopt the conventional crossed "H" structures as the metal patches and the width of each rod is t = 10 μm, as illustrated by Fig. 2(a). The symmetrical structure guarantees that the cross coupling between the x-and y-polarized EMWs can be eliminated efficiently. Besides, this structure also has an advantage in that the resonant phase under the illumination of x-and ypolarized EMWs can be independently controlled. After determining the structures of the unit cells, we next obtain the relations between the resonant phase and the structure parameters of the unit cells with the help of FEM simulation. Compared with the conventional design process, there are more restricted requirements on the unit cells in this work. The patches can not be too large or it will affect the rotation and cause the mutual coupling between the adjacent unit cells. Under this condition, by sweeping the key structure parameters, we find that the unit cells can still provide a sufficiently large resonant phase range to design the coupling metasurface. We choose six unit cells as a super cell and the exact structure parameters of the unit cells are illustrated in Tab.1. The reflection amplitude and phase versus frequency under x-polarized EMW illumination are plotted in Figs. 3(a), and 3(b), from which we can find that the high reflection efficiencies and linear phase gradients are both realized at 0.3 THz. Figure 2(b) shows the reflection phase of each unit cell under the illumination of x-and y-polarized EMWs. Δφ, which approximatively equal to 180º for all unit cells, denote the difference between φxx and φyy. This approximate constant guarantees that rxx=-ryy.

B. DESIGN OF THE PROPAGATING METASURFACE
Next, we will design the propagating metasurface so that the excited surface waves can be guided out efficiently. The propagating metasurface is periodic in the x-and y-directions with the period p = 233 μm. In addition to the wave vector match condition, to further improve the converting efficiency, the unit cells of the propagating metasurface should also be designed similar to those of the coupling metasurface in shape. In this way, the mode mismatch between the two parts can be efficiently weakened and a smooth and efficient transition between the excited surface waves and SSP can be realized. Based on the targets mentioned above, the unit cells which meet all the requirements are illustrated by Fig. 4(a). Similar to those of the coupling metasurface, the unit cells of the propagating metasurface are tri-layer structures consisting of flat copper mirror and patches, with the same dielectric embedded between them. Each patch is composed of two vertically crossed "H" structures and the width of each rod is t = 10 μm.   should cross at the frequency f = 0.3 THz with the desired longitudinal wave vector, as illustrated by Fig. 4(b). As can be seen from the dispersion relations, the curve of TM SSP overlaps with light line in the beginning and finally deviates from it while the curve of the TE SSP origins from 0.295 THz and is relatively flat in the whole frequency band. In the same range of kx, the frequency band of TM SSP is far wider than that of the TE SSP.

A. SIMULATION RESULTS OF UNIDIRECTIONAL SSP EXCITATION AND ANOMALOUS REFLECTION
With both the coupling metasurface and propagating metasurface designed, finally, we demonstrate the functionalities of CP-controlled SSP excitation and anomalous reflection. Figure 6(a) is a schematic view of the complete structure of the meta-coupler. A Gaussian beam with waist-width w = 1.2 λ is used as the incident plane wave source. For the RCP incidence, the incident waves "see" a small phase gradient (kx = 0.24 k0) and are anomalously reflected with the reflection angle θr = 13.9º. The scattering electric field components in the x-and y-directions are illustrated by Figs. 6(b), and (c), which demonstrate the function of anomalous reflection. Comparing Ex with Ey, we can also find that the chirality of the incident waves changes.
For the LCP incidence, a large phase gradient (kx = 1.2 k0) will be added on the incident waves, so the surface waves will be excited on the coupling metasurface. Due to the  matches in longitudinal wave vector and electric field modes, the excited surface waves can be efficiently guided out from the coupling metasurface to the propagating metasurface, as illustrated by Figs. 6(d), and (e). The visual electric field distributions are in agreement with theory, confirming the function of efficient SSP excitation. Next, we will quantitatively demonstrate the working performance of the designed meta-coupler.

B. CONVERTING EFFICIENCIES OF THE BIFUNCTIONAL META-COUPLER
The most important indicator to evaluate the performance of the meta-coupler is the converting efficiency. In this work, due to the difference of the scattering electric field distributions of the polarization-controlled functionalities, the converting efficiencies for the LCP and RCP incidence are calculated in different methods. For the RCP incidence, the reflected waves in the far field and the incident waves are both Gaussian traveling waves. Therefore, the square of the ratio of reflected waves to incident waves, i.e. |Er| 2 / |Ei| 2 (Er and Ei are the respective electric field amplitude of the reflected and incident Gaussian waves), can be used to evaluate the efficiency [33]. The far-field patterns of the incident and reflected waves at 0.3 THz are plotted in Fig.  7(b). In this way, the converting efficiencies versus frequency are calculated and plotted in Fig. 7(c), and the maximum converting efficiency can reach 82% at the central frequency 0.3 THz. The gray shadow in Fig. 7(c) represents the frequency band in which the converting efficiencies are over 50%. The half-power bandwidth of the bifunctional meta-coupler for the RCP incidence is 0.0285 THz. For the LCP incidence, the ratio of the power carried by SSP to that incident on the coupling metasurface, is considered to evaluate the converting efficiency [29]. The integral of poynting vector is used to calculate the power carried by SSP and the "I" in Fig. 7(a) represents the integral position. The relations between the converting efficiencies and frequency are plotted in Fig. 7(c) and marked in blue. From the efficiency curve we can know that the meta-coupler maintains high efficiency in a relatively broad frequency band and the maximum converting efficiency can reach 70% at 0.3 THz. The frequency band with efficiency higher than 50% is marked with red shadow and the corresponding halfpower bandwidth is 0.01 THz.

C. THE EFFECT OF GAUSSIAN BEAM WAIST-WIDTH
Having demonstrated the high efficiency of the designed meta-coupler, next we will study its working performance with different beam waist-widths which are defined as w.
The inset in Fig. 7(d) illustrates the definition of waist-width. All the results in last section are calculated under the condition w = 1.2 λ. The converting efficiencies versus the beam waist-width w are calculated at f = 0.3 THz, as illustrated by Fig. 7(d). In the range from 0.7 λ to 1.7λ, the bifunctional meta-coupler can maintain relatively high efficiency (>48%) for both the LCP and RCP incidence. In this range, the efficiencies of anomalous reflection have negative correlation with w while the relations are inverse for SSP excitation. This is because, for the anomalous reflection, the higher efficiency requires more EMWs impinging on the coupling metasurface. So a wider Gaussian beam always corresponds to a lower converting efficiency. However, for the SSP excitation, the higher efficiency requires more uniform incident electric field amplitude distribution on the coupling metasurface. So a relatively wider Gaussian beam always corresponds to a higher converting efficiency. When the waist-width w is out of this range, there will be evident drops in converting efficiencies.

V. CONCLUSION
This paper proposes a CP-controlled bifunctional metacoupler for independent SSP excitation and beam steering. The normally incident waves are converted into surface waves and anomalously reflected under the illumination of the LCP and RCP waves, respectively. The respective converting efficiency can reach 82% and 70% for the RCP and LCP incidence and their corresponding half-power bandwidth is 0.0285 and 0.01 THz. By elaborately adjusting the resonant phase and rotation angle of each unit cell, many other diversified functionalities can also be integrated into the SSP meta-coupler. Our work can provide another degree of freedom in polarization-controlled unidirectional excitation of SSP and a new way of encoding the polarization information in different electric field modes.