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Abstract
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 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 right or left 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 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.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces are 2D metamaterials capable of manipulating electromagnetic (EM) waves at will by imparting space-variant phase changes on the incident waves. Owing to the unique characteristics of subwavelength thickness and strong ability in wavefront control, metasurfaces have shown many potential applications in beam focusing [1–6], multichannel reflection [7–10], waveplates [11–15], holograms [16–19], vortex generation [20], and polarimeters [21–23]. In particular, by adjusting the phase gradient appropriately, metasurfaces can act as bridges linking the traveling waves and the surface waves [24]. In addition, by periodically arranging the unit cells, metasurfaces can be transformed to spoof surface plasmon (SSP) waveguides [25]. The corresponding SSP modes and dispersion characteristics can be tuned by elaborately modulating the propagating unit cells. This property increases the flexibility in designing SSP waveguides. Moreover, compared with other SSP couplers, metasurfaces also have advantages in miniaturization because the thickness is far smaller than the wavelength in free space. Various SSP meta-couplers (couplers based on metasurfaces) have been designed and fabricated in the past few years [24–28].
Recently, polarization-controlled bifunctional SSP and surface plasmon (SP) meta-couplers are proposed and experimentally demonstrated in the microwave and optical regimes [29–31]. These works open up new routes for multi-freedom-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 [32]. 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 [33]. 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 polarization-controlled unidirectional SP excitation and beam steering at normal incidence [34]. Similarly, polarization-switchable multi-functional metasurface for focusing and SP excitation are also designed by Ling et.al [35]. All these works realize the independent SSP/SP excitation and beam steering under orthogonal linear polarization electromagnetic wave (EMW) illumination. However, for the CP incident waves which have many practical applications in plasmonics and microwave systems, multifunctional meta-couplers beyond unidirectional SSP excitation, to the best of our knowledge, have not been demonstrated because of spin coupling between the orthogonal incident waves [36,37].
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 propagating phase with the Pancharatnam–Berry phase, efficient decoupling can be realized between the orthogonal CP incident waves [38–40]. 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 = 13.9° under the illumination of the left circular polarization (LCP) or the right circular polarization (RCP) beam, respectively. Finite element method (FEM) full-wave simulations performed by COMSOL Multiphysics 5.3a 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 waist-width 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 meta-coupler in this way. Besides, our work provides a new way of encoding the polarization information in different electric field modes.
Fig. 1 Schematic view of the bifunctional meta-coupler that can realize circular polarization-controlled unidirectional SSP excitation and anomalous reflection. The meta-coupler consists of a coupling metasurface and a propagation metasurface.
2. Theory analysis
We now describe the physical mechanism of spin-decoupling between the orthogonal CPs. First of all, we analyze the electromagnetic properties of the unit cells of the metasurface. Modeling the reflective metasurface as a one-port device, its reflection properties can be characterized by a Jones matrice . The coordinate base of the matrice R represents the unit vectors of the orthogonal linear polarization EMWs. If we consider the unit vectors of circular polarized waves as the new coordinate base, the new reflective Jones matricecan be expressed as
To simplify the complexity of further analysis, we design the unit cells with mirror symmetry structures. This can efficiently eliminate the cross coupling between the orthogonal linear polarization incident waves, i.e. rxy = ryx = 0 [37]. In this case, the Jones matricecan be written aswhere 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, we getwhere represents the Jones matrice of the unit cell with a rotation angle θ under the circular polarized mode base, and represents the transposed conjugation of the rotation operator . Assuming that the reflection coefficient rxx and ryy meet the relation rxx = -ryy, then the Jones matrice can be simplified asFor the CP incident waves with opposite handness, i.e. and , the respective reflected waves are and . Compared with the incident waves, we can find that 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) [36,37]. To break the symmetry and realize the independent phase control for the 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 = ejφ, which means that each unit cell has a specified propagating 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. 2(d). In this way, another propagating phase φ is introduced as a new degree of freedom. Therefore, Eq. (4) can be revised to
Fig. 2 (a) Schematic of the unit cell of coupling metasurface. The period and thickness of the unit cell are 233μm and 50μm, respectively. The exact parameters of the unit cells used in this work are listed in Table 1 and their respective amplitude and reflection phase versus frequency under the illumination of x-polarized EMWs are plotted in (b) and (d). The exact reflection phase for x- and y-polarized incidence are plotted in (c), where Δφ = |φx − φy| approximately equals to 180° for all unit cells to guarantee that rxx = -ryy.
Therefore the reflected waves for the LCP and RCP incidence can be expressed as and, respectively. By elaborately adjusting the propagating phase φ and the rotation angle θ, arbitrary independent reflection phase can be obtained for the orthogonal CP wave incidence.
3. Overall design of the meta-coupler
3.1. Design of the coupling metasurface
To realize CP-controlled SSP excitation and beam steering, 2D arrays of unit cells with different propagating 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 = 60° and Δθ = 20° so the total reflection phase difference between the adjacent coupling unit cells for the LCP and RCP incidence is 100° and 20°, respectively. Δφ and Δθ represent the phase difference of the propagating phase and Pancharatnam–Berry phase in the adjacent unit cells, 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 the reflected waves is kr = (Δφ-2Δθ) / p = 0.24 k0. Therefore, the normally incident waves are anomalously reflected with the reflection angle θr = arcsin (kr / k0) = 13.9°. For the LCP incidence, the longitudinal wave vector of the reflected waves is kr = (Δφ + 2Δθ) / p = 1.2 k0, which means that the surface waves are excited on the coupling metasurface. According to the mode coupling theory [41], the smaller the longitudinal wave vector difference, the higher the coupling efficiency. By designing a propagating metasurface which has the same longitudinal wave vector with the surface waves on the coupling metasurface, the excited surface waves can be guided out with high efficiency. When we decrease the period p, the longitudinal wave vectors of the reflected waves under orthogonal circular polarization will increase proportionally. 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 mirrors separated by a 50 μm-thick dielectric spacer Taconic RF-43 with the relative permittivity εr = 4.3 − 0.01i. This dielectric has the advantage of low loss, which can be used as low loss alternative to conventional FR-4 materials. The metal patches and mirrors are composed of copper with a thickness of 2 μm. The permittivity of copper is characterized by a Drude model ε(ω) = 1 − ωp2∕ (ω2 + iωγ), where ω is the angular frequency, ωp = 1.123 × 1016 Hz is the plasma frequency and γ = 1.379 × 1013 Hz is the collision frequency [42,43]. 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). This structure is widely used in circular polarization metasurface. 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 propagating phase under the illumination of x- and y-polarized EMWs can be independently controlled. It is noted that, the working frequency should be far from the resonant absorption band or the coupling metasurface will become a meta-absorber [44]. After determining the structures of the unit cells, we next obtain the relations between the propagating 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 demands on the unit cells in this work. The patches cannot 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 propagating 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 Table 1. The number of unit cells n in a super cell can be any figure larger than two. A larger n always corresponds to a better effect because it can form a smoother phase gradient. However, more unit cells in a super cell will lead to a larger propagating phase range, which will increase the difficulty of design. The reflection phase and amplitude versus frequency under x-polarized EMW illumination are plotted in Figs. 2(b) and 2(d), from which we can find that the linear phase gradients and high reflection efficiencies are both realized at 0.3 THz. Figure 2(c) shows the reflection phase of each unit cell under the illumination of x- and y-polarized EMWs. Δφ, which approximatively equal to 180° for all unit cells, denote the difference between φxx and φyy. This approximate constant guarantees that rxx = − ryy.
Table 1. Exact Geometric Parameters of the Units Cells unit:μm
3.2. Design of the propagating metasurface
Next, we 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. 3(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. By adding Floquet boundary condition in the x-direction, and periodic boundary condition in the y-condition, and absorption boundary condition in the z-direction on the unit cell, the Eigen frequency can be calculated by the Eigen frequency solver of COMSOL Multiphysics 5.3a. In order to meet the wave vector match condition, the dispersion curves of TE and TM modes should cross at the frequency f = 0.3 THz with the desired longitudinal wave vector, as illustrated by Fig. 3(c). To achieve this, we can adjust the lengths of the rods. The optimal structure parameters of unit cells are bx = 95 μm, lx = 133 μm, by = 107 μm, and ly = 150 μm. Besides, the Q-factor which can reflect the effective transmission distance and the frequency bandwidth of the SSP waveguide is also calculated and plotted as an inset in Fig. 3(c). The propagating metasurface composed of this unit cell supports both the TE and TM mode SSP. The electric current and electric field distributions of the TE and TM mode SSP are plotted in Figs. 3(b) and (d), respectively. As can be seen from the dispersion relations, the curve of TM SSP overlaps with the 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.
Fig. 3 (a) Schematic of the unit cells of the propagation metasurface. The length of the four rods are bx = 95 μm, lx = 133 μm, by = 107 μm, and ly = 150 μm. The period and thickness of the unit cell are 233μm and 50μm, respectively. The electric current and electric field distributions of the TE and TM mode SSP are plotted in (b) and (d), respectively. The dispersion curve (c) of the TE and TM mode SSP cross at f = 0.3 THz with the longitudinal wave vector kx = 0.55π / p. The inset in (c) illustrates the Q-factor of the coupling metasurface versus the loss tangent tanδ of the dielectric spacer.
4. Performance of the meta-coupler
4.1. 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. The overall simulations are performed by COMSOL Multiphysics 5.3a. The copper patches and mirrors are defined by Drude model and the dielectric spacer Taconic RF-43 is defined with εr = 4.3 − 0.01i, σ = 0, and µr = 1. These simulation setups are consistent with the real scenario. The wave source which can radiate the circular polarization Gaussian plane wave is placed right above the coupling metasurface with the distance of 2λ. To simulate infinite space, perfect match layers are set around the meta-coupler to absorb the scattering waves. We divide the simulation space into four parts, i.e. the air part, the dielectric part, the boundary between the air and dielectric, and the copper part. The corresponding maxium sizes of the mesh are λ/20, λ/50, λ/130, and λ/130, where λ represents the wavelength in free space. The mesh divided with high density guarantees that the results are converged and accurate. Figure 4(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 = 13.9°. The scattering electric field components in the x- and y-directions are illustrated by Figs. 4(b) and (c), which demonstrate the function of anomalous reflection. 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. 4(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.
Fig. 4 (a) Schematic view of the complete structure of the designed bifunctional meta-coupler. (b) and (c) are respective scattering electric field components in the x- and y-directions under the illumination of RCP waves. The RCP incident waves are anomalously reflected with the reflection angle θr = 13.9°. (d) and (e) are electric field components in the x- and y-directions under the illumination of the LCP waves. Surface waves are excited on the coupling metasurface and guided out from the propagating metasurface. The black dotted line represents the boundary between the coupling metasurface and propagating metasurface regions.
4.2. 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 RCP and LCP 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 [34]. The far-field patterns of the incident and the reflected waves at 0.3 THz are plotted in Fig. 5(b). In this way, the converting efficiencies versus frequency are calculated and plotted in Fig. 5(c), and the maximum converting efficiency can reach 82% at the central frequency 0.3 THz. The gray shadow in Fig. 5(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. 5(a) represents the integral position. The relations between the converting efficiencies and frequency are plotted in Fig. 5(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 half-power bandwidth is 0.01 THz.
Fig. 5 (a) Schematic of SSP excitation under the LCP illumination, where “I” represents the integral position of power of SSP. (b) The far-field patterns of the incident and reflected waves for the RCP incidence. (c) The converting efficiencies of the LCP and RCP incidence under the condition that the waist-width w = 1.2 λ, which are plotted by blue and red curves, respectively. The different shadow colors represent the corresponding frequency intervals that the excitation efficiencies are over 50%. (d) The relations between the converting efficiencies and incident Gaussian beam waist-width w. The inset illustrates the definition of waist-width. In the range from 0.7 λ to 1.7λ, the bifunctional metasurface can maintain high efficiency (>48%) for both the LCP and RCP incidence.
4.3. 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. 5(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. 5(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. Therefore, 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.
5. Summary
This paper proposes a CP-controlled bifunctional meta-coupler for independent SSP excitation and beam steering. The normally incident waves are converted into the surface waves or anomalously reflected under the illumination of the LCP or RCP waves, respectively. The respective converting efficiency can reach 70% and 82% for the LCP and RCP incidence and their corresponding half-power bandwidth is 0.01 and 0.0285 THz. By elaborately adjusting the propagating 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.
Funding
National Natural Science Foundation of China (61531002).
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