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Abstract
This work theoretically proposed dual terahertz (THz) slow light plateaus by tuning the destructive interference between a toroidal magnetic momentum and magnetic dipole momentum. The metasurfaces are in a sandwich structure. A metallic cut-wire is patterned on one side of polyimide thin-film, and a rectangular split-ring resonator (SRR) on the other side with asymmetric layout. By translating the SRR along the cut-wire from the top terminal to the bottom terminal of the cut-wire, dual slow light plateaus are found in the transparency window at a certain range of displacement. A maximum of 40.4 ps group delay is achieved as the displacement achieves 9 μm. The numerical mapping of electromagnetic field indicates that the electrical dipole on metallic cut-wire results in a localized toroidal magnetic momentum, while the inductive-capacitor oscillation of SRR results in a magnetic dipole momentum. These two momentums have opposite directions, which will repel each other at certain displacement, creating the transparency windows. Furthermore, an electrical coupling takes place in between the bilayer metasurface so that the slow light achieves a maximum, with the aforementioned two mechanisms working in coincidence.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In optics, the slow light device becomes a potential approach to develop computers that will use only a fraction of the energy of today's machines [1,2]. Simultaneously, the rapid development of computing technology not only requires low energy consuming but also calls for the high-speed device working at the frequency band from gigahertz (GHz = 109 Hz) to terahertz (THz = 1012 Hz) [3]. Therefore, an innovation on THz slow light device will be significant for the next generation of computing technology. The emergency of metasurface of plasmon-induced transparency (PIT) attracts much attention owing to its feasibility to achieve slow light cost-effectively in a broad electromagnetic frequency region [4–11]. Such a PIT effect originates from a destructive interference of intrinsic mode of basic resonators on metasurface as long as the coupled resonators have relatively close oscillation amplitude at the same resonance frequency [3–17]. A giant dispersion within this transparency window is created so as to slow down dramatically the group velocity of propagating THz pulse. Conventionally, the group delay of THz pulse is at the time scale below 30 ps by changing the excitation pathway of near-field coupling or conductively coupling in between the super-radiant and sub-radiant resonators of monolayer metasurface [11–25]. By manipulating the destructive interference of excited and induced spoof surface plasmon, a 40 ps group delay is foreseen theoretically at certain THz frequency domain [26]. However, it needs a complex design of circular cavity with 48 zigzag grooves to support the spoof surface plasmon, which could be a challenge for device integration.
For monolayer metasurface, only transverse electromagnetic components could be induced on resonators. The radiation symmetry results in a theoretical upper limit to the transmission [27]. Therefore, it is incapable of improving manipulation efficiency of transmitted light by using monolayer metasurface. However, such limit tends to be transcended by breaking the radiation symmetry with a deep sub-wavelength metasurface. To a bilayer metasurface of V-shaped resonators, transverse electric and magnetic current are supported on the top and bottom layers consisting of structures with subwavelength lateral sizes, respectively [28]. Thus, the bilayer metasurface results in discontinuities on both transverse magnetic field and transverse electric field, respectively. By breaking the radiation symmetry with structural asymmetry, a full control of transmitted light with higher efficiency becomes possible. In the bilayer metasurfaces, strong intra-layer couplings among adjacent unit cells should be taken into account due to their extremely small lateral distance. Furthermore, the separation between two layers is at deep subwavelength scale (below one-tenth of the operation wavelength), which also leads to strong inter-layer couplings. Owing to these two couplings, higher manipulation efficiency of transmission manipulation, relative to monolayer metasurfaces, must be attributed to the coupling effects [29,30]. Since 2010, the bilayer metasurfaces have being exhibited higher efficiency than the monolayer metasurface in manipulating electromagnetic radiation, such as polarization conversion, chirality control, and quality factor enhancement [31–44].
In this work, we propose a novel approach to maximize the THz slow light effect of the group delay up to 40 ps. Such a bilayer asymmetric metasurface is in sandwich structure. It consists of one metasurface of periodic metal cut-wire, and another one of rectangular split-ring resonators (SRR). The symmetric axis of SRR is shift away from the cut-wire so that the two metasurfaces are asymmetric in xy plane. An ultrathin layer of polyimide is in between the two metasurface. We provide an analytical and numerical insight into the evolution of the induced transparency windows as well as the localized slow light phenomenon with displacement of SRR translating along the cut-wires.
2. Metasurfaces pattern and simulation methods
Figure 1 presents the schematic diagram of the hybrid structure in a unit cell of bilayer asymmetric metasurface. The unit cell is based on a 2.5 μm thick polyimide thin-film of 125 μm × 125 μm, which is patterned with a layer of metallic cut-wire and another layer of rectangular SRR on both sides, respectively. On the top layer of metasurface, the cut-wire is 98 μm in length and 5 μm in width as well as 0.2 μm in thickness. The cut-wire is in the middle of the unit-cell along the Y-axis. To the bottom layer of metasurface, the rectangular SRR is placed on the backside of the polyimide thin-film, which is shift away from the cut-wire 19.5 μm along the x-direction. Thus, the second layer of SRR become asymmetric metasurface to the first layer of cut-wire. The SRR has 32 μm long arm with one 6 μm gap. The thickness and width of SRR are 5 μm and 6 μm respectively. Our structure is similar to the earlier work [45], however, the evolution of transparency windows is totally different. Until today, it is a technical challenge to make patterns on both-sides of 2.5 μm thick polyimide thin-film. The surface tension results in wrinkles as long as the thickness of film is thin enough. The thinnest commercially available polyimide thin-film is about 25 μm thick, which is too much thick to achieve interlayer coupling. As such, this work focuses on the numerically simulation excluding the experiment. The simulation results are obtained from a FDTD algorithm-based platform CST Microwave Studio. The time-domain solver is adopted with the unit-cell boundary conditions in the x-y plane of 125 μm × 125 μm square area. The input and output ports along the z direction are set 30 μm away from the top layer metasurface and bottom layer metasurface, respectively. The permittivity of polyimide is ε = 3.5, of which the loss tangent is 0.0027. The electric conductivity of gold is σ = 4.6 × 107 S/m. Since we use time domain solver for simulation, an experimental THz waveform is better than the default signal of CST package. Herein, a time-domain THz pulse signal from a THz-time domain spectrometer (Menlosystem Tera K15) is used as excitation source, of which its temporal window is 17 ps. Correspondingly, the temporal interval between two point is 1/3 ps. The simulation frequency range is from 0.2 THz to 2 THz. The transmittance T(ν) as well as the phase spectrum of the bilayer asymmetric metasurface can be calculated by the functions of S-parameters T(ν) = |S21|2. The mesh cell is hexahedral for simulation. The totally mesh cell is 30250. The mesh density is 50 lines per wavelength.
Fig. 1 (a) Schematic diagram of THz radiating on the bilayer asymmetric metasurface, KTHz refers to the wavevector of incident THz pulse. ETHz and HTHz refer to the electrical components and magnetic components respectively. (b) The sandwich structure of bilayer metasurface is 125 μm × 125 μm, in which L = 98 μm, D = 2.5 μm, t = 0.2 μm, w1 = 5 μm, w2 = 6 μm, a = 32 μm, g = 6 μm, respectively.
The THz transmittances of a single cut-wire and an individual SRR are shown in Fig. 2(a), which are derived from the fast Fourier transform of the measured THz time domain data. One Lorenzian lineshape resonance occurs in THz spectrum when the THz polarization is parallel to the cut-wire along the Y-axis; however, the other Lorenzian lineshape resonance mode occurs to the SRR only the THz polarization is along the X-axis. The Q factors of resonance modes of aforementioned two types of resonators are calculated as below [35]:
where ν is the mode frequency and Δν is the mode linewidth. Correspondingly, the details of resonance modes are listed in Table 1.Fig. 2 (a) THz transmittance of cut-wire and of SRR under THz irradiation with different polarization. Blue solid-line refers to the transmittance of cut-wire. Red solid-line refers to the transmittance of SRR. The electric density at resonance modes of (b) cut-wire and of (c) SRR, respectively. The surface current distribution at resonance mode of (d) cut-wire and of (e) SRR, respectively. Color bar: the relative strength of electrical field as well as the surface currents.
Table 1. The mode frequency of basic resonators
To achieve PIT effect, it is required that the mode-frequencies of two basic resonators are nearly identical but with quality factors (Q factors) in contrast, namely a narrow sub-radiant mode and a broad super-radiant mode. Table 1 indicates that the Q factor of resonance mode of individual SRR is much higher than that of the cut-wire. At this point, the cut-wire can be recognized as super-radiant bright resonators, and the SRR plays the role as sub-radiant dark resonators. It is evident that the central frequencies of both types of basic resonators are identical. As such, the detuning factor of resonance frequency of super-radiant and sub-radiant resonators can be neglected. The damping ratios of intrinsic modes of cut-wire and SRR are close since the linewidths of both modes are similar. The electric energy distributions of intrinsic modes of both resonators are shown in Figs. 2(b)-2(c), both of which are accumulating at the terminals of the metal structures. To the SRR, there is obvious energy localization at the gap area. Figures 2(d)-2(e) show that the mono-directional surface currents of surface plasmon oscillation dominate the super-radiant mode of cut-wire, and a circulating current flow along the inner-edge of the SRR, which is the evidence of inductive-capacitive (LC) resonance. Then, the feasibility of PIT effect from bilayer asymmetric metasurface need to be revealed.
3. Results and discussion
The simulated THz transmittance as a function of frequency versus displacement δ of SRR is presented in Fig. 2 correspondingly. In the case of incident THz polarization being perpendicular to the gap of SRR (EX), there are two resonance modes appear in THz spectrum without any displacement of SRR (δ = 0 μm), as shown in Fig. 2(a). With the δ increasing up to 8 μm, these two modes overlap to be one mode. Then, a further increasing of δ will separate these two modes more far away in frequency domain. The spectral profile in between the two modes is as flat as the frequency background, which is not the evidence of the transparency windows. Therefore, we stop to discuss such a case. In the case of incident THz polarization being parallel to the cut-wire (EY), there is a medium transparent within a narrow spectral range around a transmittance minimum area in the spectrum (δ = 0 μm), as shown in Fig. 2(b). The central frequency of such a transparency window is termed as νT. Interestingly, this transparency window closes when the displacement δ increases from 0 μm to 16 μm. The frequency of the transmission minimum νC is almost the same as νT. Alternatively, another transparency window opens when the δ is beyond 16 μm, and the width of the second transparency window increases monotonically with the δ increasing. The central frequencies of both transparency windows are identical. Since the destructive interference between the intrinsic modes of basic resonators leads to the PIT, the central frequency of transparency window should overlap with the basic resonators. In our case, the νT is about 1.18 THz the same as the intrinsic modes of both resonators shown in Fig. 2(a). A much finer simulation of the THz transmittance of MM as a function of δ and frequency are illustrated in Fig. 3(c), in which the simulated step of δ is 1 μm. In agreement with the data shown in Fig. 3(b), two peaks of transmission minimum move closer one another with the δ increasing from 0 μm to 15 μm, and merge to a mono-resonance mode νC at δ = 16 μm. To a further increasing of displacement (16 μm < δ < 66 μm), the resonance mode νC is separated, which give rise to a second induced transparency window of νT, and it becomes wider and wider monotonically.
Fig. 3 (a) The THz transmittance of bilayer asymmetric metasurface with X-polarized THz incidence. (b) The THz transmittance of bilayer asymmetric metasurface with Y-polarized THz incidence. The δ is the displacement of SRR to the cut-wire. (c) 2D map of THz transmittance as a function of frequency and displacement with a finer step of 1 μm.
According to our previous results, a PIT-like behavior can mimic the transparency windows of PIT by tuning the dual modes of resonators into a very close distance in frequency domain; however, the slow light is invisible in the fake transparency window [14]. Therefore, the influence of slow light is a key criterion to evaluate the PIT effect, which is a positive group delay (Δτ) at the transparency window in spectrum [15–17]. Here, the Δτ represents the time delay of THz wave packet instead of the group index. The Δτ can be calculated from the equation as below [8]:
where φ and ν refer to the effective phase and frequency of THz complex transmission spectrum, respectively. Figure 3(a) shows the simulated phase spectrum of the bilayer asymmetric metasurface, in which distinct phase transitions are found at two adjacent frameworks (transmission minima) of the transparency windows. The extracted group delays as a function of THz frequency of bilayer asymmetric metasurfaces are illustrated in Fig. 4(b). With the δ increasing from 0 to 8 μm, the Δτ increases monotonically from 28.3 ps to 38.6 ps, however, this value reduces to 0 when the δ is at 16 μm. When the δ increases up to 24, the Δτ achieves 19.5 ps. A finer map of group delay is simulated as a function of frequency and δ in Fig. 4(c), in which the simulation step of δ is 1 μm. The two frameworks of the transparency windows exhibit negative group delay, which is the same as the previous results of PIT effect [15–17]. However, two plateaus of positive group delay occur in the transparency windows. The first slow light plateau appears in the range from δ = 0 μm to δ = 15 μm, where the maximum of Δτ reaches 40.4 ps at δ = 9 μm. The second slow light plateau appears in the range from δ = 18 μm to δ = 50 μm, where the peak of Δτ reaches 33 ps at δ = 27 μm. The occurrence of slow light plateau is the evidence of localization of the PIT effect. Although the two plateaus of slow light possess the same central frequency, the plateau shape and the peak value of THz slow light at above two transparency windows are not the same. Therefore, the strength of destructive interference of resonance modes of cut-wire and SRR need to be analyzed with a deep insight at two transparency windows.Fig. 4 (a) The phase spectra and (b) the group delay of bilayer asymmetric metasurface with displacement δ of SRR from 0 to 64 μm at the interval of 8 μm. (b) The two-dimensional diagram of group delay as a function of frequency and displacement δ. The step of displacement δ is 1 μm.
In order to reveal the origin and evolution of dual transparency window νT, the electrical energy densities of bilayer asymmetric metasurfaces are simulated numerically at the frequencies of νT and νC correspondingly, as shown in Fig. 5. Herein, we select the bilayer asymmetric metasurfaces of three different displacements for simulation. The first case is δ = 8 μm, the first transparency window of a relatively larger slow light plateau. The second case is δ = 16 μm, where the transparency window closed completely. The third case is δ = 24 μm, where the second transparency window νT opens with a relatively smaller slow light plateau. In the first case, the incident THz wave give rise to a uni-directional surface current from upside to downside on the cut-wire, which results in an anti-clockwise current flow on the SRR. According to the Ampère's right-hand grip rule, a cylindrical magnetic field that wraps round the cut-wire is generated. Biot-Savart law can extract the magnetic field [46,47]:
where dl is a vector along the path C whose magnitude is the length of the differential element of the wire in the direction of conventional current. The μ0 = 4π × 10−7 H/m is the permeability of vacuum. The rd is the full displacement vector from the wire element (dl) to the point at which the field is being computed (r). Such an induced cylindrical magnetic field passes through the enclosed area of SRR. The Lenz’s raw indicate that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field. Thus, another generated magnetic field with opposite polarity takes place inside the SRR enclosed area. As is shown in Fig. 5(b), a strong magnetic field pass through the enclosed area of SRR resulting in anti-clockwise current on the SRR. Here, the magnetic field from cut-wire is termed as B1, and that from SRR is termed as B2. The cylindrical B1 is a symbol of magnetic toroidic dipole moment m1, which pass through the enclose area of SRR with opposite direction to the B2. The B2 refers to another magnetic dipole momentum m2. The magnetic momentum is defined as below [46,47]:Here, the Br is the magnetic field strength at the position of r, V is the volume of the magnet. The magnetic dipole–dipole interaction between m1 and m2 leads to a potential vector H. Mathematically, the H is as below [46,47]:Thus, the interference between opposing magnetic dipole minimizes the potential vector, which gives rise to the transparency windows in THz frequency domain. At δ = 16 μm, the intrinsic magnetic dipole m2 disappears so that the potential vector H achieve maximum, which refers to the individual resonance νC. Thus, the transparency window closes. In the third case, the surface currents on the cut-wire exhibit localization phenomenon: its strength become stronger on the section below the gap of SRR, and the direction of current flow maintains on the entire cut-wire. As such, the intrinsic mode of cut-wire shifts slightly away to the high frequency component. Meanwhile, the anti-clockwise current flow on the SRR becomes much weaker than the first case. Normally, the shorter current on cut-wire refers to a relatively higher resonance frequency. The resonance frequency of intrinsic LC loop obeys the rules as below [46,47]:At δ = 24 μm, the surface current accumulates more on the bottom arm than on the both sides of the gap of SRR, as shown in Fig. 5(a). As a result, this resonance becomes more inductive rather than capacitive so that the central frequency of LC mode shifts slightly away to the low frequency component. Thus, the spatial localization of surface current on cut-wire and SRR leads to a frequency shift slightly toward the opposite direction. As a consequence, the second transparency window become wider and wider with the displacement δ increasing. Simultaneously, such a frequency deviation between the two intrinsic modes of cut-wire and SRR leads to an amplitude reduction of destructive interaction owing to the super-position principle of wave interference. Therefore, the dispersion of the second transparency windows is relatively smaller than that of the first transparency windows. In agreement to the Fig. 5(b), the m1 and m2 in the first transparency window is supposed to be stronger than those in the second window. Then, the destructive interference of magnetic dipole-dipole interaction is much stronger. As such, the slow light in the first transparency window is larger than in the second window.Fig. 5 (a) Surface currents distribution as well as (b) the diagram of magnetic field of THz transparency windows of bilayer asymmetric metasurface at δ = 8 μm, 16 μm, and 24 μm, correspondingly. The νT and νc refer to the transparent window and closed windows respectively. The color bar refers to the relative strength of surface currents.
Furthermore, the inductance coupling is not the only mechanism for above dual transparency windows phenomenon. In a two dimensional metasurface, the sub-radiant mode resonance is able to be excited by both the electric and magnetic fields when the SRR translates along the cut-wire without altering the lateral distance between the resonators [48]. To a three dimensional system of bilayer metasurface, however, the metal resonators as well as polyimide substrate works like a capacitor. As such, the interlayer electrical coupling from the top layer metasurface of cut-wire to the bottom layer metasurface of SRR has to be taken in account. Since the incident THz polarization is along the Y-axis, only the project at the xz-plane of electrical field need to be discussed. Thus, the x component and z component need to be considered together for the electrically interlayer coupling:
Here, Er is the electrical field at the position r(x,z) in space. Ex is the x component, and Ez is the z component. As it can be seen from the simulated fields in Fig. 6, the Ex on the edge of cut-wire has the opposite polarity to the gap area of SRR at δ = 8 μm and δ = 24 μm. The strongest electric field concentration locates in the gap area of the SRR on back layer. As such, the sub-radiant mode of LC resonance is excited sufficiently owing to the electrical coupling. At δ = 16 μm, the electric field concentrates on the two terminals of the cut-wire, which has the same polarity to the electrical field distribution in the gap of SRR. Meanwhile, the electrical energy in the gap area of SRR is not as strong as the other two case (δ = 8 μm and δ = 24 μm). It can be elucidated that the LC resonance of SRR is not sufficiently be excited at δ = 16 μm. In comparison, the Ez focus mainly on the two side of the top arm of SRR. Correspondingly, its electrical field polarity at δ = 8 μm and δ = 24 μm are opposite to that at δ = 16 μm. As such, we believe that the Ex play the key roles in the strong electric field coupling from the wire edge to the SRR pair. The surface currents on cut-wire and SRR with δ = 8 μm and δ = 24 μm excited by the Ex and Ez are oscillating in phase. The constructive interference between these two fields leads to a strong excitation of the LC resonance in the SRRs, enhancing the coupling between the super-radiant and sub-radiant modes. However, when the SRRs move to δ = 16 μm, the surface current on cut-wire remains constant but the electrically excited surface current as well as the electrical fields on SRR become very weak, as shown in Fig. 5 and Fig. 6, respectively. Thus, magnetic dipole-dipole interact destructively to cancel each other so as to form the two slow light plateaus at induce transparency windows. At δ = 16 μm, the electrical coupling counteracts the magnetic coupling, which leads to a total coupling nearly zero that eventually leads to the disappearance of the transparency window.Fig. 6 Electrical field distribution of THz transparency windows of bilayer asymmetric metasurface at δ = 8 μm, δ = 16 μm, and δ = 24 μm, correspondingly. The νT and νc refer to the transparent window and closed windows respectively. EX refers to the electrical component along x-axis, EZ refers to the electrical component along Z-axis. The color bar refers to the relative strength of electrical field.
4. Summary
In bilayer asymmetric metasurface made of cut-wire and SRR in sandwich structure, a phenomenon on dual THz slow light plateaus at transparency windows is predicted via numerical simulation. By tuning the displacement of the SRR translates along the cut-wire without altering the lateral distance between the resonators from up to the bottom, the transparency window appears to be an open-close-open behavior. A maximum of 40.4 ps group delay is foreseen in the first localized THz slow light plateau. The second slow light plateau has a relatively lower maximum of group delay. The central frequency of both windows are almost the same. The numerical mapping of electromagnetic field indicates that a destructive inference occurs in between the excited magnetic toroidic dipole momentum and induced magnetic dipole momentum, which results in the transparency window. Simultaneously, a capacitive coupling in between the bilayer asymmetric metasurface achieve maximum, which enforces the slow light effect in transparency windows.
Funding
Joint Research Fund in Astronomy (U1631112) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS).
Acknowledgments
Zhenyu Zhao and Zhidong Gu contribute equally in this work so that both are lead authors.
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