Coupling-based Huygens' meta-atom utilizing bilayer - OSA Publishing

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Jun 30, 2017 - meta-atom by coupling a nanodisk to its Babinet-complementary structure ..... in weak capacitive coupling which is inversely proportional to td.
Vol. 25, No. 14 | 10 Jul 2017 | OPTICS EXPRESS 16332

Coupling-based Huygens’ meta-atom utilizing bilayer complementary plasmonic structure for light manipulation TONGJUN LIU,1 LIRONG HUANG,1,* WEI HONG,1 YONGHONG LING,1 JING LUAN,1 YALI SUN,1 AND WEIHUA SUN2 1Wuhan

National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China 2Wuhan Maritime Communication Research Institute, Wuhan, Hubei, 430079, China *[email protected]

Abstract: Huygens’ meta-atom is the basic building unit of Huygens’ metasurfaces allowing for almost arbitrary wavefront shaping across a surface. We here present a kind of Huygens’ meta-atom by coupling a nanodisk to its Babinet-complementary structure (nanohole), and develop an optical lumped nanocircuit model to analyze vertical and lateral coupling effects and resonance frequencies. Simulation results show that the tuned coupling via lateral misalignment between the two nanostructures is sufficient to shape the wavefront without changing the dimensions or orientations of antennas. By tuning the coupling via lateral misalignment, we design a reflective gradient metasurface based on one coupled mode and a high-efficiency transmissive gradient metasurface working in the spectral overlap of electric and magnetic resonances to realize beam deflection. The proposed coupling-based Huygens’ meta-atom is a new building block for plasmonic metasurfaces with enhanced light-matter interactions, high-efficiency and almost arbitrary wavefront shaping over the full electromagnetic spectrum. © 2017 Optical Society of America OCIS codes: (160.3918) Metamaterials; (160.1245) Artificially engineered materials; (260.5740) Resonance; (050.5080) Phase shift; (350.4238) Nanophotonics and photonic crystals; (260.2110) Electromagnetic optics.

References and links 1.

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#294526 Journal © 2017

https://doi.org/10.1364/OE.25.016332 Received 25 Apr 2017; revised 28 Jun 2017; accepted 28 Jun 2017; published 30 Jun 2017

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1. Introduction Huygens’ metasurfaces, also known as equivalent impedance metasurface [1–5], composed of arrays of Huygens’ meta-atoms (or Huygens’ sources) represented by orthogonal electric and magnetic polarization currents, allow for almost arbitrary field distributions on both sides of a surface. These subwavelength Huygens’ meta-atoms can mimic the emission of secondary wavelets as proposed in the Huygens’ principle or equivalence principle, and hence allow for perfect control on the wavefronts of electromagnetic waves and almost arbitrary wavefront shaping across a surface [1–3]. Since a Huygens’ metasurface possesses both electric and magnetic responses, it can also realize highly directive beam [6], high-transmission wavefront manipulation with negligible reflection and absorption loss [7–9], independent control on the magnitude and phase of locally reflected or transmitted wave [10, 11], and impedance-matching with environment to boost transmission or absorption efficiency [1,7,12]. Previously demonstrated plasmonic Huygens’ meta-atoms obtained by cascading multilayer patterned metal sheets exhibit high-efficiency wavefront manipulation in the microwave regime [1,11,13], however, they are less appealing in the optical regime due to fabrication difficulty, the unavailability of lumped circuit, high energy dissipation in metals [14], and the lack of suitable magnetic materials at optical frequencies. Instead, implementations of “Huygens’ like” source based on nanocircuit paradigm at optical and infrared frequencies have adopted a multilayered structure [3,15], which has relatively lower efficiency due to the intrinsic loss of metallic materials and also imposes a challenge on its application and fabrication because of its stacking design. Moreover, for an ideal dielectric Huygens’ source, a complete phase coverage from 0 to 2π needs an exact spectral overlapping of electric and magnetic resonances [7,16], hence setting a limit on its operation bandwidth. In addition, the inherent magnetic saturation problem in the visible range makes it very hard to achieve plasmonic Huygens’ source over the full spectrum [9, 17]. Bearing these challenges in mind will motivate us to find an ideal path towards attaining total control on optical wavefronts. We here present and demonstrate a new route for constructing a Huygens’ meta-atom by coupling a nanodisk to its Babinet-inverted structure (nanohole). This kind of bilayer complementary structure, naturally possessing electric and effective magnetic responses over the entire electromagnetic spectrum according to Babinet’s law [18], is a potential Huygens’ source for constructing Huygens’ metasurfaces with enhanced light-matter interactions and high-efficiency wavefront shaping. It was initially used for compact microwave pass-filters [19,20], but recently has attracted great attention in the optical regime due to its giant coupling effect in improving the performances of nanophotonic devices including high extinction ratio polarizers [21], high-transmission cross-polarization components [22], and high-resolution coloration products [23], etc. The strong coupling effects inherently existing in the bilayer structures can provide strong light-matter interactions, enhanced nonlinear effects, giant optical activity, and multi-dimensional control on electromagnetic waves, therefore having significant advantages in designing nanophotonic devices with improved performances [24,25]. Moreover, noticeably different from other’s works on phase gradient metasurfaces [26–33], where phase discontinuity is achieved by changing either the geometrical dimensions or orientation angles

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of meta-atoms, we demonstrate that the tuned coupling via misalignment between the two nanostructures is sufficient to shape wavefronts without changing the dimensions or orientation angles. To verify the capability, we design a reflective gradient metasurface working in the low-frequency coupled mode of this meta-atom for abnormal beam reflection. In addition, to highlight the merit of Huygens’ meta-atom, a high-transmittance Huygens’ metasurface working in the spectral overlap of electric and magnetic resonances for transmitted light-wave manipulation is also conceived. This paper is organized as follows. In section 2, we illustrate the proposed Huygens’ metaatom and its resonance mechanism; in section 3, we develop an optical lumped nanocircuit model to systematically investigate the vertical and lateral couplings and the resonant frequencies of the proposed Huygens’ meta-atom, and then discuss the validity of this model by employing numerical simulations. Furthermore, to verify the tunability of the proposed Huygens’ meta-atom, two kinds of gradient metasurfaces, which respectively work in abnormal reflection and abnormal refraction modes, are designed in section 4, and finally a conclusion is given in section 5. 2. Configurations and resonance mechanism of Huygens’ meta-atom Figure 1(a) depicts the proposed bilayer complementary Huygens’ meta-atom with periods along the x-axis and y-axis L = 800 nm. It consists of a silver (Ag) nanodisk with radius r = 150 nm and thickness t = 30 nm in the top layer, and its complement, i.e. a nanohole, in the bottom layer. They are separated by a silica dielectric layer with thickness td. The misalignment of the nanodisk with respect to the nanohole along the x-axis is denoted as s. When illuminating the nanodisk with an x-polarized light, a localized surface plasmons resonance with an electric dipole (ED) moment along the x-axis can be excited, leading to a pronounced dip in the transmission spectrum [34]. According to Babinet’s principle, and given not too high dissipative loss, several quantities in a complementary system exchange their respective roles: transmission and reflection, electric and magnetic field, as well as the two orthogonal polarization modes [18,34]. Therefore, the nanohole in free space, under the illumination of an x-polarized light, exhibits a dominant magnetic dipole (MD) resonance along the y-axis which causes a peak in the transmission spectrum [18,34].

Fig. 1. (a) Schematic view of the designed Huygens’ meta-atom. (b) Transmission spectra for nanodisk and nanohole. (c) The corresponding near-field distributions (z-component of electric field Ez for nanodisk and z-component of magnetic field Hz for nanohole) in the x-y plane for the respective resonance peaks in (b).

To investigate the performance of the meta-atom, full three-dimensional (3D) finitedifference-time-domain (FDTD) simulations are carried out by employing the FDTD solver from Lumerical, Inc. An x-polarized plane wave normally illuminates the structure from the negative z-axis. Periodic boundary conditions are applied in the x–y plane, and the perfectmatch layer condition is used along the z-axis. We first simulate the individual nanodisk and

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the individual nanohole arrays with period L = 800 nm. As shown in Fig. 1(b), a sharp dip (point A) at wavelength 0.94 μm can be observed for the nanodisk array, indicating that the nanodisk array serves as a band-stop filter; in contrast, a sharp transmission peak (point B) can be also seen around 1.01 μm, showing that the nanohole array behaves as a band-pass filter. One may wonder why these two nanostructures do not exhibit the same resonance wavelengths. To address this, it should be stressed that the precondition of the Babinet’s law is that the metal should be perfect electric conductor [18]. In our case, the metal is silver, where Babinet’s law is not rigorously satisfied. In fact, in our structures, this deviation is mainly due to the area ratio of the metal, which is defined as the ratio between the area of metal and that of a meta-atom, and will affect the effective plasma frequencies of the studied nanodisk and nanohole structures [17]. To understand the nature of the excited modes, we examine the normal component (the zcomponent) of the electric and magnetic near-field distributions as shown in Fig. 1(c), which clearly shows that the nanodisk exhibits an electric dipole p oscillating in the x-direction when illuminated by an x-polarized light (see the electric field distributions in the upper panel of Fig. 1(c)). The resonance is therefore associated with an electric dipole lying within the sample plane. In contrast, as shown in the lower panel of Fig. 1(c), the nanohole exhibits a dipolar resonance in the y-direction which has magnetic characteristic as a result of the magnetic moment m created by enhanced electric field-induced ring current flowing in the equatorial plane around the aperture [18,34]. The resonance is therefore associated with a magnetic dipole lying within the sample plane. It is important to note, however, that the picture of a magnetic dipole is only a reformulation of electric fields which ultimately mediate the coupling in the system. Thus, the electric and virtual magnetic dipole responses are confirmed in the nanodisk and nanohole arrays respectively, just as predicted by Babinet’s principle. Now we discuss the case when the nanodisk is coupled with the nanohole to form a Huygens’ meta-atom. It has been demonstrated highly symmetric structures such as nanodisks and nanoholes are efficient and straightforward to couple [34]. Figure 2 shows the simulated amplitude and phase responses of the transmitted and reflected light from the designed Huygens’ meta-atom with the silica dielectric thickness td respectively equals 100 and 20 nm (s is fixed as 0 nm here). Figures 2(a)–2(c) are for the case when td = 100 nm, what is intriguing is that there are four resonant peaks instead of one at wavelengths 0.83 μm, 0.86 μm, 1.0 μm and 1.53 μm in the transmission spectrum in Fig. 2(a). Note that although peaks at 0.83, 0.86 and 1.0 μm with limited transmittance are close to each other and not easily to distinguish, but with the aid of phase change as plotted by the red line in Fig. 2(a), we can still discern them, because resonances are always accompanied by phase changes [35]. Further investigation by examining the corresponding electric charge distributions (see the electric field distributions in Fig. 2(c)) reveals that these four peaks respectively correspond to higher-frequency coupled mode (hfcm), ED mode for the nanodisk, MD mode for the nanohole, and lower-frequency coupled mode (lfcm). As can be seen from the second and third subfigures of Fig. 2(c), electric charges mainly concentrate in the upper nanodisk or lower nanohole layer, hence, we can readily infer that the two peaks at 0.86 μm and 1.0 μm derive from ED mode and MD mode, respectively. While for the two coupled modes, electric charges concentrate in both the nanodisk layer and nanohole layer as shown in Fig. 2 (c), indicating they are resulted from the interactions between the two nanostructures. As for Huygens’ meta-atom, 2π phase coverage is vital for wavefront controlling. Although, our designed Huygens’ metasurfaces for reflection and transmission work in lfcm and the overlapping region of ED and MD modes (which will be discussed later in Section 4), we here only focus on the transmitted and reflected phase responses of this structure for lfcm shown as the red lines in Fig. 2. We first discuss the case when td = 100 nm, from Figs. 2(a) and 2(b) we can see that the transmitted/reflected fields show a maximum phase change of π across the corresponding resonance wavelength for lfcm. This π phase change is insufficient to fully

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control light, however, 2π phase coverage can be indeed achieved by further enhanced coupling via reducing td.

Fig. 2. Simulated results for the electric-field amplitude (blue) and phase (red) of the transmitted light (a) and the reflected light (b) when td = 100 nm. Green dashed lines denote the location of peaks. (c) The corresponding electric intensity patterns in the x-z plane for the respective resonances in (a). The white line marks the simulated structure. (d) Simulated results for the electric-field amplitude (blue) and phase (red) of the transmitted light and (e) the reflected light when td = 20 nm.

Then, we turn to the case when td = 20 nm. In contrast to the case td = 100 nm, we can observe from Fig. 2(e) that the phase response of the reflected field changes drastically and undergoes a phase change of 2π for the lfcm. It should be pointed out here that the dielectric thickness td we choose later to construct Huygens’ metasurfaces for abnormal light reflection is just 20 nm. Meanwhile, it cannot be ignored that only π phase change happens in the transmitted field of lfcm, which is due to the asymmetric structure of the designed Huygens’ meta-atom and hence the electromagnetic response behaves quite differently in the upper reflection half-space and the lower refraction half-space [22]. The two coupled modes can be qualitatively explained from hybridization model proposed in ref. 36. The high-frequency coupled mode (hfcm) is characterized by in-phase charge oscillations in the nanodisk and the nanohole, which form symmetrically arranged dipoles that repulse each other, and hence gives rise to an enhanced interaction that results in a higher resonance frequency. By contrast, for the low-frequency coupled mode (lfcm), the corresponding charge oscillations are out-of-phase, and the anti-symmetrically arranged dipoles attract each other, hence decreasing the restoring force and leading to a lower resonance

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frequency [36]. Simple and straightforward analysis about the two coupled modes will be further given in the following section, where we first develop an analytic optical lumped nanocircuit model to analyze coupling effects and resonance frequencies, and then we discuss the validity of this analytic model by carrying out full three-dimensional FDTD simulations. 3. Vertical and lateral couplings in the Huygens’ meta-atom 3.1 Analytical optical lumped nanocircuit model In order to gain an intuitive understanding of the coupling effects in the proposed Huygens’ meta-atom, we develop an analytical nanocircuit model, which is built on the concept of metamaterial-inspired optical circuitry [37–39]. A collection of nanostructures may form a ‘circuit of light’, which, when excited by an optical signal, shows local electric fields and displacement currents ∂D/∂t analogous to voltage and current distributions in conventional radio-frequency circuits. A nanostructure may function as a lumped circuit element, such as a nanocapacitor, a nanoinductor or a nanoresistor, provided that its permittivity satisfies Re(ε) > 0, Re(ε) < 0 or Im(ε) ≠ 0, respectively. Here, however, we focus on the resonance frequencies of two coupled modes and note that the loss of the metal associated with nanoresistors does not influence the resonance frequency, so the presence of material loss is not a concern. Thus, metal and dielectric (air gaps included) in our case can be reasonably treated as nanoinductors and nanocapacitors, respectively. When the Huygens’ meta-atom is illuminated by an x-polarized optical signal, the optical displacement current I1, I2, I3 ‘flows’ along the three layers of the meta-atom, respectively. Analytical lossless optical lumped nanocircuit model for the Huygens’ meta-atom thus can be shown as Fig. 3 which includes three parallel parts representing the top nanodisk layer, middle dielectric layer and bottom nanohole layer, respectively. For the nanodisk array illuminated by an x-polarized optical signal, the displacement current ‘flows’ transversely across the nanodisks and air gaps. Thus, this structure functions as a lumped nanoinductor Lnanodisk and nanocapacitor Cnanodisk in series, which behaves as a bandstop filter as discussed in section 2. However, under the same illumination, the nanohole array works as a band-pass filter, and it can be represented by a parallel combination of lumped nanoinductor Lnanohole and nanocapacitor Cnanohole [37]. The coupling between the two structures is represented by a shunt nanocapacitor C determined by the dielectric layer and mutual inductance coupling coefficient M depending on the misalignment of two structures.

Fig. 3. Analytical optical lumped nanocircuits model for Huygens’ meta-atom.

The capacitance and inductance for the nanodisk array can be simply estimated using the equations Cnanodisk = ε0εeffS0/L and Lnanodisk = µ0πr2/t. Here, S0 = 2rt is the projected area of the nanodisk in the x-z plane, εeff is the effective dielectric constant of the dielectric environment for the nanodisk, which is simply and straightforward taken as 1 when we calculate the resonance wavelength of the nanodisk in the vacuum. Then the resonance wavelength of the nanodisk in the vacuum (εeff = 1) is calculated as λ = 2πc(LnanodiskCnanodisk)1/2 = 1.02 µm, which

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approximates the simulated result, 0.94 µm as shown in Fig. 1(b). According to Babinet’s law, the capacitance and inductance of the nanohole in the vacuum can then be calculated by Cnanohole = 4ε0Lnanodisk/µ0 and Lnanohole = LnanodiskCnanodisk/Cnanohole [40]. As for the equivalent capacitance C for the middle dielectric layer, it mainly contributes to the capacitive coupling in the Huygens’ meta-atom, and can be estimated by C = ε0εSiO2Seff/td. Here, Seff is the effective area of the equivalent nanocapacitor, which represents the effective interaction area between the nanodisk and nanohole layers. We roughly choose the area of nanodisk/nanohole to estimate the effective area Seff, i.e., Seff = πr2. Later discussion in Section 3.2 will indicate that the calculated results for meta-atoms are qualitatively consistent with the simulated ones, which manifests this approximate method to estimate Seff is effective. When discussing the coupling between the nanodisk and nanohole arrays separated by silica layer from the circuit model, it also needs to address that εeff, the effective dielectric constant of the dielectric environment for the nanodisk and nanohole, should take the permittivity value between the vacuum and silica when they are put on silica dielectric layer with varied thicknesses. Here, however, we simply and straightforwardly use the vacuum permittivity (εeff = 1) in the analytical model because in this model we have treated the designed meta-atom as three individual parts, i.e., nanodisk in the vacuum, nanohole in the vacuum, and silica dielectric spacer. According to Kirchhoff’s voltage law, the directed sum of the electrical potential difference (voltage) around any closed network is zero, thus we get  1 1 + iω Lnanodisk +  iωC  iωCnanodisk  1 −  iωC    −iω Lnanohole 



1 − iω Lnanodisk M iωC



0

1 1 + iωC iωCnanohole



1 iωCnanohole (1)

iω Lnanohole +



1 iωCnanohole 1 iωCnanohole

   I  0  1      I2  =  0 .  I  0  3     

where ω is the angular frequency. Resonant frequencies for lfcm and hfcm can be derived under the condition that the determinant of the matrix of coefficients equals zero. For the lfcm, the displacement currents I1, I3 ‘flow’ reversely in the nanodisk and nanohole layers which means the electric charges distribute anti-symmetrically, hence resulting in additive magnetic field coupling. While for the hfcm, the electric charges distribute symmetrically, therefore I1 and I3 flow in the same directions, resulting a subtractive magnetic field coupling. The symmetric and antisymmetric displacement current flows lead to a resonance splitting in the Huygens’ metaatom. Using Eq. (1) and εeff = 1, the resonance wavelengths of the two coupled modes with varied dielectric thickness td and mutual inductance coefficient M are obtained and given in Fig. 4, wherein lfcm and hfcm, respectively, are marked by a cluster of dotted and solid lines. As can be clearly shown by the two black arrows in Fig. 4, with the increase of td, the two coupled modes gradually move towards the eigen ED and MD modes (denoted by pink area) of the two discrete nanostructures. Besides, for td = 50, 200 and 400 nm and 0.1< M < 0.6, the two coupled modes merge into one mode, this is because the dielectric thickness td is larger and thus it results in weak capacitive coupling which is inversely proportional to td. In addition, the inductive coupling is also not strong enough due to small inductive coefficient M. So, the total coupling strength is not sufficient to split lfcm and hfcm, which makes lfcm and hfcm not resonate and merge into the ED/MD mode. When td is further increased beyond one certain value (200 nm in our case), lfcm and hfcm both display small frequency shift against increasing td, which means coupling effects saturate. Besides, when td increases and hence capacitive coupling weakens, it requires more inductive coupling to split the resonance.

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As a matter of fact, it should be remembered that the effect of dielectric thickness td is mainly responsible for the vertical capacitive coupling in the Huygens’ meta-atom. Now we discuss the lateral inductive coupling which are mainly quantified by mutual inductance coefficient M. With the increase of M, the resonance wavelength of lfcm experiences blueshift at first, and then redshift (whereas hfcm behaves in the opposite trend), these are the consequence of the competition between capacitive and inductive couplings.

Fig. 4. Numerically calculated resonance wavelengths from Eq. (1) for lfcm and hfcm with varied dielectric thickness td and mutual inductance coupling coefficient M. lfcm and hfcm, respectively, are marked by dotted and solid lines. The resonance spectral range for electric dipole (ED) and magnetic dipole (MD) are represented by pink area.

Besides, we also calculate the case when td equals 100 nm and M takes 1.0 (not shown here), the calculated resonance wavelengths for the two coupled modes are 0.3 μm and 1.15 μm, respectively, which deviate from the corresponding simulated results in Fig. 2, which are 0.83 and 1.53 μm, respectively. The discrepancy mainly lies in the smaller εeff we take. In fact, when we take a larger εeff such as 1.6 (note εeff has a permittivity value between those of vacuum and silica which depends on the thickness of the dielectric layer), the calculated result for lfcm is 1.50 μm, approaching the simulated one of 1.53 μm. However, for the hfcm, the calculated result still deviates from the simulated one to a large extent, we guess the larger discrepancy is partly because Fabry-Pérot effect becomes stronger for higher frequencies and larger thickness td [41]. In addition, the roughly estimated value for Seff, and the non-local feature of highfrequency resonance mode [42] may also be responsible for the larger discrepancy. Last but not least, we treat the metal of nanodisk and nanohole as perfect electric conductor, which ultimately limits the applicability of this analytical model to some extent. Therefore, we have to admit that this circuit model is not suitable for accurate calculation for large td and short-wavelength hfcm resonances. However, it still can qualitatively predict the varying trend of resonance wavelengths for hfcm and lfcm, which may shed some light on designing metasurfaces. On the other hand, one can design metasurfaces working in different modes depending on different capacitive and inductive couplings. For example, when capacitive coupling is stronger (td is smaller) or inductive couplings is stronger (M is bigger), the gap between the resonance wavelengths of hfcm and lfcm is relatively larger, then we can use this to design metasurfaces working in hfcm or lfcm resonance. In contrast, when td is big and M is small, capacitive and inductive couplings are both weak, lfcm and hfcm merge into the ED/MD mode marked by the pink area in Fig. 4. These overlapping regions can be used to design high-transmission Huygens’s metasurface operating near the frequency where MD and ED resonances coincide, which will be discussed in Section 4.

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3.2 Numerical simulation results To discuss the validity of the optical lumped nanocircuit model, we numerically simulate the resonance responses of the Huygens’ meta-atom. We first focus on the vertical capacitive coupling by varying dielectric layer thickness td. Figure 5 shows the reflectance spectral evolution as td varies from 20 nm to 420 nm while keeping s = 0 nm. The dotted lines are used to intuitively show the evolution trends of various modes. The type of modes were judged by observing the electric field distribution at resonance peaks, in the same way we discern the mode types as shown in Fig. 2. Then, by finding the minimal reflectance at a certain dielectric layer thickness td, we locate the corresponding resonance wavelengths. After repeating the above processes for different td, we finally draw the dotted lines to show the various mode trends in Fig. 5. Grating-assisted excitation of surface plasmon polariton (SPP) at the metal−dielectric interface is observed attributing to the lattice/period of the Huygens’ metaatom. The positions of the SPP resonances depend on the grating-induced change in momentum parallel to the surface and are therefore highly dependent on the structure period and effective dielectric permittivity. In addition, the gap between hfcm and lfcm is rather wide for very small td as a result of stronger capacitive coupling effect between the two complementary metallic layers. But as td increases, the gap becomes narrower with hfcm red-shifting while lfcm blueshifting, thereby making the two coupled modes gradually move toward the eigen MD and ED resonances of the nanodisk and the nanohole. What’s more, as td increases, frequency shift becomes less noticeable as a result of the weakened capacitive coupling. It can also be found when td ranges from 118 to 152 nm, the MD and ED are overlapped with zero reflection, indicating the occurrence of impedance matching. When dielectric thickness td increases beyond 200 nm, we can find little frequency shift for lfcm which means coupling effects saturate. These simulated results are qualitatively consistent with the calculated ones in Fig. 4. However, unlike the analytical circuit model predicts, no overlap happens for the four modes although they come closer to each other. This is mainly because Fabry-Pérot effect [41] gradually emerges when increasing the thickness of dielectric layer, however, the circuit model does not consider Fabry-Pérot effect.

Fig. 5. Reflectance spectral evolution for Huygens’ meta-atoms when varying dielectric thickness.

In the above, we mainly focus on the vertical capacitive coupling between the upper nanodisk layer and the lower nanohole layer, which is dependent on the dielectric layer between them, especially on the dielectric thickness td. We now turn to discuss the lateral coupling mechanism by setting a misalignment s between the upper nanodisk and bottom nanohole layers. The dielectric thickness td is fixed at 20 nm for obtaining strong coupling effect in the Huygens’ meta-atom. Since the period of the meta-atoms is 800 nm, we only need to change s from 0 to 400 nm. Simulated reflectance and phase responses are mapped in Fig. 6. As can be seen in Fig. 6(a), a blueshift followed by a redshift occurs for lfcm with the increase of lateral shift s while the opposite trend holds for hfcm (when s is larger, hfcm overlaps with ED mode),

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this is because lateral misalignment s mainly influences the mutual inductance in the Huygens’ meta-atoms and hence inductive coupling. Inductive coupling competes with capacitive coupling and thus causes total coupling strength to decrease at first and then strengthen. From another perspective, coupling strength depends on the interaction between two plasmonic resonances, namely the overlap of the driven electric charges. When s equals 0 or 400 nm, the electric charge profiles of the two plasmonic resonances have the largest spatial overlap resulting in strongest lateral coupling, thus the two coupled modes both exhibit maximal frequency shifts with respect to the resonant frequencies of the two individual nanostructures. These results are also qualitatively consistent with the corresponding analytical results about the effect of the mutual inductance in Fig. 4. Discrepancies between the calculated and the simulated resonance frequencies for lfcm and hfcm mainly lie in the effective dielectric constant is more than 1 for the realistic nanodisks and nanoholes, however, it takes 1 when calculating these two modes according to Eq. (1). In addition, the roughly estimated value for Seff also brings in discrepancy. Therefore, the analytical circuit model is only approximate at best. Like most circuit models, it fails to calculate the amplitude and phase changes of resonances although it is simple and straightforward to qualitatively predict resonance frequencies.

Fig. 6. Evolution of the simulated reflectance (a) and transmittance (c) spectra, and phase responses for reflection (b) and transmission (d) with varied misalignment s. The black squares denote the overlapping region of ED and MD.

We now focus on the phase response of lfcm when tuning the inductive and effective capacitive coupling via varied misalignment s. As can be seen in Fig. 6(b), reflectance phase change from 0 to 2π can be obtained by tuning misalignments s over a broad spectral region (marked by two vertical white dashed lines in Fig. 6(b)) because the equivalent impedance of the Huygens’ meta-atoms changes from capacitive reactance to inductive reactance. This 2π phase coverage is significantly important because almost arbitrary spatial distribution of light can be generated by designing the phase profile. And in the following section, we will employ this to design a reflective Huygens’ metasurface for wavefront manipulation. Furthermore, the spectral overlap of ED and MD is observed around s = 380 nm with near zero reflection and high transmittance as shown by the black rectangle in Figs. 6(a) and 6(c), this is a result of out-of-phase superposition of the two dipoles. From the phase response of transmitted light in Fig. 6(d), we can recognize that the overlap of electric and magnetic dipoles (marked by black rectangle) can realize 2π phase coverage, which can be used to accomplish

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Huygens’ metasurface for wavefront shaping as reported in [7–9]. In the next section, we will employ this to design a high-transmission Huygens’ metasurface to realize light bending. 4. Huygens’ metasurface for light bending As discussed above, 2π phase coverage, which is important for wavefront shaping and light manipulation, can be achieved by laterally tuning misalignment s and the coupling between the nanodisk and nanohole. Here, we employ the Huygens’ meta-atoms to design a Huygens’ metasurface, which is able to bend normal incident light to an anomalous reflection angle by imposing a phase gradient dφ(x)/dx along the interface. According to the generalized Snell’s law [26], for a surface with phase gradient along the interface, the refracted and reflected beams follow

λ dϕ ( x) nt sin (θt ) − ni sin (θi ) = 2π dx λ dϕ ( x) ni sin (θ r ) − ni sin (θi ) = . 2π dx

(2)

where θi, θt, and θr are the angles of incidence, refraction, and reflection, respectively; λ is the optical wavelength in vacuum. And ni and nt are the refractive index of the reflected and transmitted half-space, which both equal 1 in our case. It should be stressed that generalized Snell’s law approach is approximate at best and fails for large angles. There are fundamental limitations in passive gradient metasurfaces for molding the impinging wave, and local phase compensation is essentially insufficient to realize arbitrary and efficient wavefront manipulation, hence full-wave designs should be employed [43–45]. We first focus on the lfcm and then turn to the frequency range where ED and MD nearly coincide. In fact, many previous works have focused on the overlap between the ED and MD modes to realize Huygens’ metasurfaces [7–9], we here employ lfcm for a new scheme to design Huygens’ metasurface. It is comprised of an array of periodically arranged Huygens’ metaatoms with varied misalignments s as shown in Fig. 7. In our case, we choose four meta-atoms to compose a supercell (region surrounded by red dashed line) with period Γ = 4L along the xaxis. The four discrete meta-atoms have different misalignments s (s = 60, 100, 120, 360 nm, respectively), offering the reflected beam with almost equal amplitude responses and π/2 phase increments to cover 0 to 2π phase, thereby realizing full phase control on the reflected wavefront. It is worth mentioning that the four meta-atoms have the same structure except for their difference in misalignments s.

Fig. 7. Schematic view of the designed Huygens’ metasurface.

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Figures 8(a) and 8(b) respectively depict the simulated scattered wavefronts of amplitude and phase. When a linearly x-polarized plane wave at wavelength 1.4 μm normally impinges on the metasurface, the reflected light is deflected at an oblique angle θr almost 26° as shown in Figs. 8(a) and 8(b), which is in agreement with the value (25.9°) calculated from the generalized Snell’s law in Eq. (2). While in the transmitted half-space, the incident light-wave are mainly refracted into the normal transmittance. The wavelength-dependent efficiency for the Huygens’ metasurface is shown in Fig. 8(c), from which we can find more than 61% of the incident energy is transferred into the anomalous reflection order while nearly 36% transmits into the transmission order with less absorption around 1.4 μm. Compared with the reflection efficiency in refs. 29 and 30 where metallic back-mirror are used, the reflection efficiency here is not so high because part of the incident energy leaks into the transmitted mode. However, our designed metasurface may serve as a planar ultrathin power splitter working in the anomalous reflection and normal transmitted orders. What’s more, operation bandwidth is also a key factor for metasurfaces, we map the normalized intensity of the reflected field as functions of wavelength λ and reflection angle θr in Fig. 8(c). As can be seen clearly, the Huygens’ metasurface can work well in abnormal reflection mode in wavelength range from 1.35 to 1.49 μm. It should also be noted that in spite of the abovementioned features, the two-layer design may potentially limit the angular operability of this device.

Fig. 8. Scattered electric field amplitude (a) and (b) phase distributions for the designed metasurface under normal incidence. The reflection angle θr is almost 26°. The white dashed lines denote where the metasurface locates. (c) Reflectance (R), transmittance (T) and absorption (A) spectra. (d) Normalized intensity of the reflected field as functions of wavelength and reflected angle θr. The intensity is normalized to the maximal intensity over the whole range, which means the maximum of the two- dimensional matrix for the color map is 1.

The merit of Huygens’ metasurface lies in its ability to realize high-efficiency transmitted light manipulation. Our meta-atoms can act as Huygens’ meta-atoms with high transmission

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and controllable phase near the frequency where the MD and ED resonances almost coincide, its operation principle is similar to those reported in references [7–9]. As we have discussed in Section 3.2, the black rectangles in Figs. 6(a) and 6(c) indicate one can realize hightransmission phase-shifting device by tuning the misalignment s between nanodisk and nanohole. We therefore carefully select a set of misalignment s (s = 333, 356, 380, 400 nm) to realize high transmission Huygens’ metasurface at wavelength 0.88 µm. The simulated wave fronts for the scattered fields and reflectance/transmittance/absorption spectra are shown in Fig. 9. When a linearly x-polarized plane wave at wavelength 0.88 μm normally impinges on the metasurface, the refracted light is bent into an oblique refraction angle θt (almost 16°) as shown in Figs. 9(a) and 9(b), which is in agreement with the calculated value (15.9°) using the generalized Snell’s law in Eq. (2). we can find the normal incident beam is efficiently coupled into the abnormal refraction order with transmittance of 78%, highlighting the merit of the Huygens’ metasurface. Although there are some losses and reflection preventing higher transmittance, this metasurface is relatively efficient. And the relatively distorted and imperfect wavefronts are caused by other undesired modes, indicating further design optimization should take into account the effect of variation of wave impedance to realize 100%-efficiency coupling to the desirable mode [43–45].

Fig. 9. (a) Scattered electric field amplitude and (b) phase distributions for the designed metasurface working in the ED and MD overlapping region under normal incidence. The reflection angle θr is almost 16°. The white dashed lines denote where the metasurface locates. (c) Transmittance (T), reflectance (R) and absorption (A) spectra for the designed metasurface.

It’s needed to point out that, though we do not address the fabrication of our metasurface structure, the detailed fabrication process of a similar bilayer complementary structure can be found in reference [34], it includes two-step electron beam exposure, metal evaporation, lift off, and the experimental results indicate the accurate control of the misalignment between nanostructures can be achieved, despite of the fabrication challenges. 5. Conclusion In summary, we propose a new kind of Huygens’ meta-atom (or Huygens’ source) based on the coupling effects between a nanodisk and its Babinet complementary structure nanohole, and develop an analytical optical lumped nanocircuit model to systematically investigate the vertical and lateral coupling mechanisms by considering the effects of dielectric thickness and misalignment. Numerically simulated results are qualitatively consistent with the analytical ones. Our study shows that the two coupled modes are generated by the symmetric and antisymmetric displacement currents ‘flowing’ between the two individual layers of the Huygens’ meta-atom, and the dielectric spacer layer between the nanodisk and nanohole layers mainly contributes to capacitive coupling and vertical coupling with the coupling strength inversely proportional to its thickness. In comparison, the misalignment between the two complementary layers mainly affects inductive coupling and lateral coupling. These coupling effects will lead to significant frequency shift for the two coupled modes. Besides, when

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misalignment is introduced, the inductive coupling will compete with the capacitive one and hence enable the phase response of the Huygens’ meta-atom to gradually change from 0 to 2π for light wavefront manipulation. By tuning the coupling via varying lateral misalignment s between the two nanostructures, instead of changing the dimensions or orientations of antennas as others did, we design a reflective gradient metasurface and a high-efficiency transmissive gradient metasurface for beam deflection, the former is based on the lfcm resonance mode, whereas the latter works in the spectral overlap of electric and magnetic resonances. Since the proposed Huygens’ meta-atom is based on the strong coupling effects inherently existing in the bilayer structures, we believe the concept of coupling-based Huygens’ meta-atom and metasurface will provide a new approach for strong light-matter interaction, enhanced nonlinear effect, giant optical activity, and multi-dimensional control on electromagnetic waves, therefore having significant advantages in designing nanophotonic devices with improved performances. Funding National Natural Science Foundation of China (No. 61675074).