1. Introduction

Phase control is one of metasurfaces’ marvelous abilities. Many novel functions like plasmonic flat lens [1–3], anomalous refraction and reflection [3–6], broadband light bending [7,8], optical vortex generating [6,9,10], polarization state manipulating [11,12] have been realized with elaborately designed metasurfaces. Early metasurfaces for optical phase control are all based on single layer ultrathin metallic nanostructures [1–3, 5–7, 11]. Transmitted optical phase can be manipulated efficiently by gradually changing the structures or orientations of subunit cells. However, only tangential electric currents can be sustained in these metasurfaces, thus, reflection and absorption are inevitable. As a result, their efficiencies are rather low.

In 2013, C. Pfeiffer et al. demonstrated the first metamaterial realization of Huygens’ surface which is able to realize high efficiency phase control [13]. Magnetic current as well as tangential electric current can be sustained in Huygens’ metasurfaces. The transmission efficiency of Huygens’ metasurfaces can reach near unity when magnetic current and electric current with equal strength overlap with each other [14]. High index dielectric nanoparticles exhibit very low intrinsic loss. Besides, they can support both electric and magnetic dipolar Mie-type modes simultaneously. So far, Huygens’ surfaces constructed by high index dielectric nanoparticles [9,10,14–19] have shown great efficiency enhancement in optical phase control compared to early metasurfaces. But most of these dielectric Huygens’ surfaces are built with silicon, which is lossy above 300 THz. Dielectric phase control metasurfaces at visible wavelengths are realized with materials with wider band gap like TiO₂ [20, 21]. But a wider band gap usually means a lower refractive index. The fabrication of the TiO₂ nanoparticles with high aspect ratio is rather challenging.

Metallic Huygens’ surfaces for high efficiency optical phase control are all realized with multilayer metallic nanostructures in which magnetic current can be sustained by interlayer coupling [22–24]. In microwave range, metallic materials exhibit low loss, the multilayer structures are still easy to fabricate. However, when it comes to optical frequency, metal’s ohmic loss gets prominent, moreover, fabrication of the multilayer nanostructures becomes very difficult. Even so, several multilayer metallic metasurfaces for optical phase control have already been fabricated and achieved much higher efficiencies compared with early single layer metallic phase control metasurfaces [23, 25]. Yet, the highest simulated efficiency of metallic metasurface for optical phase control is still below 40%. There is still a lot work to be done to push the state-of-art efficiency to a higher level. Metallic metasurfaces are essential for many applications like enhancing the field for quantum and nonlinear processes [26, 27], providing electrical contacts [28, 29]. Thus, exploring the practical limitations of plasmonic metasurfaces is still of great theoretical and practical significance.

In this letter, we demonstrate that single layer metallic metasurface constructed by thick nanoparticles can realize high efficiency (above 85%) optical phase control. The thick nanoparticle designed can support bulk magnetic resonance (BMR) on its sidewalls. In addition, parallel sidewalls of adjacent thick nanoparticles constitute lateral nano cavities, in which lateral Fabry-Perot resonance can be excited. Both of the two resonances exhibit high transmission with near zero reflection. The peak position of the FP resonance is mainly decided by cavity length. But the transmitted phase is very sensitive to the cavity width. By adjusting the length of thick nanoparticles, high efficiency optical phase control can be realized readily with the single layer metallic metasurfaces.

2. Structure and methods

The structure of the proposed metasurface composed of thick nanoparticle arrays is illustrated in Fig. 1. Lattice constants of the metasurface are Px = Py = 600 nm in both the x-axis and y-axis directions. The thick nanoparticle is silver nano cuboid with t = 360 nm in thickness, l = 400 nm in length, and w = 200 nm in width. The whole structure is embedded within silica, whose dielectric constant is set to be 2.15. Optical parameter of silver is taken from Rakic’s work [30]. The spectral response and field distribution of the metasurfaces are studied with finite element method (FEM) simulations using a commercial software COMSOL. Incident wave is polarized along x-axis direction and propagate against z-axis direction. The region simulated is a single unit cell with periodic boundary conditions applied to x-axis and y-axis direction, perfectly matched layers applied to z-axis direction.

 

Fig. 1 Schematic illustration of the metasurface composed of thick silver nanoparticle arrays embedded in silica. The inset shows the dimension of the thick nanoparticle. The structure parameters are l = 600 nm, w = 200 nm, t = 360 nm, Px = Py = 600 nm, respectively.

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3. Results and analyses

Figure 2 shows the transmission property in dependence on structure parameters. As can be seen from Fig. 2(a), there are clearly three different resonances. Obviously, mode C is the ± 1st order Rayleigh anomaly. It appears when the diffracted wave propagates along the metasurface plane [31, 32]. Its spectral position can be decided by the simplified relation λ = n_SilicaPy here. Spectral position of mode A is insensitive to the periodicity in y-axis direction Py, while mode B has a strong dependence on Py. Relationship between metasurfaces’ spectral responses and other structure parameters are demonstrated in Figs. 2(b)-(e). We can conclude from Fig. 2 that the spectral position of mode A is mainly decided by the size of the thick nanoparticles (l, w, and t), the periodicity in both the x-axis and y-axis directions have little effect on it. These are the characters of nanoparticle’s localized surface plasmon resonance. In fact, this mode is a bulk magnetic resonance (BMR) as will be confirmed later.

 

Fig. 2 (a) Transmission spectrum versus Py, other structure parameters are l = 400 nm, w = 200 nm, t = 360 nm, Px = 600 nm, respectively. (b)-(e) present the transmission dependence on l, w, t, and Px, initial structure parameters are l = 400 nm, w = 200 nm, t = 360 nm, Px = 600 nm, Py = 620 nm, and each parameter is tuned to find its effect on metasurfaces’ transmission property.

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However, mode B is only sensitive to the structure parameters in y-axis direction (i.e. Py and w), the particle length l, thickness t, and periodicity in x-axis direction Px have little effect on its spectral position. Thus, we can take mode B as the result of scattering coupling of incident field in y-axis direction. But its peak wavelength is beyond the ± 1st order Rayleigh anomaly, as can be seen from Fig. 2(a). The effect of increase in Py and decrease in nanoparticle width w all increase the length of the cavity formed by neighboring nano-cuboid arrays, and give rise to a redshift of mode B. Therefore, we can infer that mode B is a FP cavity resonance supported in the cavity formed by parallel sidewalls of adjacent thick nanoparticles, as will be confirmed later with field distributions related to this mode.

To find out the origin of these two modes, we analyzed the field and current distributions related to mode A and mode B. For a nano-cuboid arrays with l = 400 nm, w = 200 nm, t = 360 nm, and Px = Py = 600 nm, its spectral response is shown in Fig. 3(a). Mode A and mode B locate at 1460 nm and 1070 nm, respectively. Both of the two modes exhibit high transmission and near-zero reflection. Current distribution related to mode A (the BMR) is displayed in Fig. 3(d). Corner effect has been considered with a curvature of 10 nm at sharp edges and corners. As can be seen, strong surface current elements are mainly located at the sidewalls of the thick nanoparticles. Each current element represents a dipolar electric resonance. The two dipolar resonances on parallel sidewalls are antiparallel. Therefore, they form a bulk magnetic resonance together. As a result, reflectance and absorption of the metasurface are suppressed, the BMR mode with high transmission appears within its transmission spectrum. Additionally, antiparallel dipolar electric resonances on adjacent sidewalls of neighboring thick nanoparticles along x-axis direction can couple to each other, thus, the magnetic field in the gap between adjacent nanoparticles in x-axis direction is enhanced due to the resulted subradiant magnetic resonance, as shown in Fig. 3(e). Electric field distribution of the BMR shown in Fig. 3(f) is mainly confined to the ends of the nanoparticles, which also indicates that this mode is a localized surface plasmon resonance.

 

Fig. 3 Spectral response, field and current distributions of the metasurface with l = 400 nm, w = 200 nm, t = 360 nm, and Px = Py = 600 nm. (a) Transmittance (T), reflectance (R), and absorption (A) of the metasurface. (b)-(c) Normalized electric field distribution at 1070 nm (mode B). (d)-(f) Localized Current elements, normalized magnetic field, and normalized electric field distributions at 1460 nm (mode A), respectively.

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In fact, the dark BMR observed here cannot be excited directly under normal incidence. Common methods to excite magnetic resonance are using asymmetry structures [33, 34], plasmon hybridization effect [35, 36], and dipole source [37, 38]. However, the thick nanoparticles are identical to each other. Thus, the symmetry is not broken. Under normal incidence, the phases of incident field experienced by different sidewalls are also identical. Therefore, the plasmon hybridization effect is not involved. The BMR here can be understood by drawing analogy to the magnetic resonance in plasmon induced transparency (PIT) [38]. Dipolar electric resonance on the top surface of the thick nanoparticle can be excited by external field, and it can be taken as a dipole source. Then, the dipole source induces the BMR, which is similar to the formation of PIT. Thus, the BMR here also gives rise to a transmission peak as the PIT effect does.

The electric field distribution for mode B is shown in Fig. 3(b) and 3(c), we can see clearly that energy is confined to the nano-cavity formed by the side walls of neighboring thick nanoparticle arrays, which is just the characteristic distribution of a FP resonance. Generally speaking, optical path corresponding to cavity length should be integer times of half the resonant wavelength. However, scattering coupling of the incident field introduces an extra phase retardation. At the same time, the coupling between FP resonance and localized surface plasmon resonance also shifts FP resonance’s spectral position [39–41]. As a result, dependence of the FP resonance’s wavelength on cavity length is not a simple linear relationship, as can be seen in Fig. 2(a).

Peak position of the lateral FP resonance is hardly affected by the nanoparticle length l as demonstrated in Fig. 2(b). This is because the peak position of a FP resonance is decided by parameters relating to cavity length (w and Py) rather than cavity width (i.e. l). However, we can also observe a degeneration in transmission when nanoparticle length l decreases. This is because a shorter nanoparticle length l means a smaller scattering object and a narrower FP cavity, which will lead to a lower scattering coupling efficiency between external field and the FP cavity. As a result, a shorter nanoparticle length l induces a lower transmission peak. Longer nanoparticle length induces much efficient scattering coupling between incident field and the FP cavity, thus, can achieve much stronger confinement effect. It is well known that optical cavity with stronger confinement effect has longer photon decay time. That’s to say, more time is needed for the photons trapped in the cavity to escape. Then the phase delay induced by cavities with stronger confinement effect will be larger. Therefore, the phase delay introduced by the FP resonance can be tuned by varying nanoparticle length l.

For metasurfaces with Px = 600 nm, Py = 650 nm, w = 200 nm, t = 360 nm, lateral FP resonances locate at 1150 nm. Their optical responses versus nanoparticle length l are displayed in Fig. 4(a). When l decreases, the BMR shifts to shorter wavelength side and overlaps with the lateral FP resonance at around l = 300 nm. Highly enhanced local field distribution of the FP resonance strengthens the interaction between optical field and the nanoparticles, thus, results in a stronger BMR and magnifies the absorption loss of metallic nanoparticles, then induces a transmission dip at l = 300 nm. The structure parameters adopted here have already been optimized to get high transmission efficiency. In fact, the ohmic loss can be further suppressed by reducing particle width w. But the improvement in transmission is very slight. Besides, a smaller w means a more challenging fabrication process. Therefore, we still set w to be 200 nm to get an acceptable aspect ratio as well as high efficiency. Even so, the minimum transmission efficiency is still above 75%, and the maximum reflectance is still negligible. What’s more, relative phase delay can be tuned from 0 degree to 270 degrees by increasing l from 100 nm to 500 nm, as shown in Fig. 4(b). The pseudo-color field map in Fig. 4(c) indicates the light transmitted from each discrete thick nanoparticle arrays, clearly revealing the discrete phase shifts with a step of 0.5π. Average efficiency of the single layer metallic metasurfaces is above 85%, which is much higher than previous metallic metasurfaces for phase control in optical wavelength range.

 

Fig. 4 Optical response at 1150 nm (the lateral FP resonance peak) in dependence on nanoparticle length l, other structure parameters are w = 200 nm, t = 360 nm, Px = 600 nm, and Py = 650 nm. (a) Transmittance, reflectance, and absorption of the metasurfaces. (b) Phase delay. (c) Normalized electric field intensity Ex/|Ein|’s distribution for arrays composed of different nanoparticles, clearly revealing the discrete phase shifts.

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4. Discussion

Most of the phase control metasurfaces capable of achieving 2π phase coverage can be used to demonstrate anomalous refraction and flat-lens focusing. Metasurfaces for anomalous refraction consist of super-cells with full 2π phase coverage. It usually needs eight or more different sub cells to construct a super-cell. Phase step between neighboring sub-cells is π/4 or even smaller. A larger phase step may result in a larger refraction angle. But the performance may degenerate at the same time. The maximum phase coverage we can realize is 1.5π. That’s to say, only four sub cells can be used to get 2π phase coverage with a constant phase step of π/2. The phase discontinuity is too large. What’s more, the phase control here is realized through tuning the cavity formed by neighboring nanoparticles. It is different from previous metasurfaces in which a certain phase delay is realized with single unit cell. In this paper, a certain phase delay is realized with a certain periodic array which composes of identical nanoparticles, rather than a single unit cell. The interaction between neighboring identical nanoparticles dominates the phase response of the periodic array. A super cell with 2π phase coverage cannot be realized with the combination of different nanoparticles. Therefore, the structure proposed in this paper is not suitable for applications like beam deflecting and focusing.

For phase control metasurfaces with phase coverage of 1.5π, vortex beam generation and hologram imaging are possible applications, as demonstrated by Katie E. Chong et al. [10, 19]. They used four periodic arrays with different periodicities to get 2π phase coverage with a step of 0.5π, and obtained vortex beam and 2D hologram image. In those cases, phase control is also realized with periodic arrays rather than separate nanoparticles. It is the same for the metallic metasurfaces proposed in this paper. A 1.5π continuous phase coverage can be realized by tuning the particle length l. Four arrays can be used to achieve 2π phase coverage with a step of 0.5π, as demonstrated in Fig. 4(c). Then, high efficiency vortex beam and hologram image can also be generated with the combination of these periodic arrays in the same way. In addition, optical properties of the metallic surface is polarization dependent. For incident wave polarized along y-axis direction, the transmission is high but phase response is flat (not shown here). It is easy to get 90° and 180° phase delay between two orthogonal field components. Therefore, high efficiency waveplates [42, 43] and vector beam generation [44] are also possible applications of the metallic metasurface proposed.

5. Conclusion

In conclusion, we investigated the optical properties of metasurfaces composed of thick nanoparticle arrays embedded in silica. We found two modes exhibiting high transmission and near zero reflection. The BMR mode can be sustained on thick nanoparticles’ sidewalls, and the lateral FP resonance can be generated in the nano cavity formed by adjacent thick nanoparticles. In addition, the lateral FP resonance can be utilized to manipulate the transmitted phase efficiently. We demonstrated high efficiency phase control with the thick nanoparticle arrays. The average efficiency achieved is about 85%, which is much higher than the highest efficiency (about 40%) realized by previous metallic metasurfaces for optical phase control in transmission mode. Though not suitable for applications like beam deflecting and focusing, the method presented in this paper may still induce a series of new metasurface designs for various applications like vortex beam generation, hologram imaging, polarization converting, biosensing, optical filtering, and enhancing light-matter interaction.