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Abstract
This work numerically investigates a localized terahertz (THz) slow light phenomenon by tuning the spoof localized surface plasmon-induced transparency (PIT). A binary meta-molecule supports the interaction of the spoof localized surface plasmon (spoof-LSP), which is composed of a metallic arc and a textured circular cavity of periodic grooves. By tuning the central angle θ of the arc from 90 degrees to 170 degrees, a slow light plateau is found in the transparency window at certain frequency range. A maximum of 46 ps group delay is achieved at the θ of 135. The numerical mapping of the electromagnetic field indicates a new-born dipolar spoof-LSP that appears at the transparency windows on the circular cavity with opposite polarity to the spoof-LSP on the metallic arc. These two spoof-LSPs of opposite direction lead to a fake quadrupole, which will repel each other in magnetic dipole momentum. The slow light achieves maximum with the induced spoof-LSP and is the same as the origin spoof-LSP on the metallic arc in oscillation strength. This work paves a new way for the maximization of THz slow light.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nowadays, decelerating the light speed from near-infrared to terahertz (1 THz = 1012 Hz) band using solid-state devices becomes one of the hottest topics in photonics [1]. As a consequence, a lot of novel approaches are proposed to achieve THz slow light, such as asymmetric waveguides based on gyroelectric semiconductor [2,3], plasmonic metallic grating [4], and meta-molecules (MM) [5–7]. Among them, the MM of plasmon-induced transparency (PIT) attracts much attention owing to its feasibility to achieve slow light cost-effectively [5–27]. Such a PIT effect originates from a destructive interference of intrinsic mode of basic resonators inside MMs as long as the coupled resonators have relatively close oscillation amplitude at the same resonance frequency [8–27]. This effect creates giant dispersion within this transparency window, which slows down the group velocity dramatically of propagating THz pulse. There is a variety of basic resonators having been successfully used in designing MMs of PIT effect, such as split ring resonator (SRR), cut-wire, U-shaped resonator [15–19]. By changing the excitation pathway or tuning the distance of near-field coupling in between the aforementioned resonators, one can normally achieve a group delay of THz pulse at the time scale of 10 picoseconds (ps) [15–18]. An alternative approach to enlarge the THz group delay is to couple the basic resonators conductively, which can achieve localized slow light up to 28 ps at certain THz frequency domain [27]. In this way, however, the properties of THz slow light strongly relies on the geometric design and the layout of conductively coupled basic resonator. To our best knowledge, a 30 ps group delay remains a challenge to the PIT effect from THz MMs until today.
The PIT effect originates from the destructive inference of surface plasmons (SPs), which are coherent delocalized electron oscillations that exist at the interface between metallic layer and dielectric substrates. Normally, the SPs can be classified into two types: surface plasmon polaritons (SPPs) which are propagating modes that travel along a metal-dielectric interface and localized surface plasmons (LSPs) which are resonance modes supported by subwavelength metal particles [28-29]. To the LSPs, the electric fields near the particle’s surface are greatly enhanced and the particle’s optical absorption has a maximum at the frequency from visible to infrared region [28-29]. At regimes of the electromagnetic spectrum far lower than the plasma frequency (such as THz and microwave frequencies), metals behave as perfect electric conductors (PEC), which prevents them from supporting SP modes [30–32]. However, it is found that a periodically textured closed surface can support mimicked LSPs beyond PEC limit, namely spoof-LSPs, including multipolar spoof-LSPs, magnetic spoof-LSPs, and multiband spoof-LSPs [33–37]. These spoof-LSPs exhibit very strong energy confinement on the periodically textured closed surface, which results in sharp resonance modes. Therefore, a destructive inference of aforementioned spoof-LSPs is able to induce a transparency window [38]. Following this way, one can generate much larger dispersion for THz slow light than any other published MMs of PIT effect.
In this work, we propose a novel approach to achieve THz slow light of the group delay beyond 30 ps. Such a binary MM is in combination of one metal arc, and another periodically corrugated circular cavity with high contrast grooves on inner-edge. The former plays the role as super-radiant resonator and the latter plays the role as a sub-radiant resonator. We provide an analytical and numerical insight into the evolution of these transparency windows with different central angle of metal arc.
2. Meta-molecule pattern and simulation methods
Figure 1 presents the schematic diagram of the hybrid structure of a single unit cell of MM. The circular cavity of periodically corrugated surface has an inner-edge with radius R = 152 μm, which is overlaid with inward radial groove of height h = 60 μm. The N = 48 is the total number of grooves, and inner radius of open area is r = 90 μm. The radial grooves are filled with a dielectric of refractive index ng, and the medium in the inner cylinder with radius r is assumed to be air. The metal arc is concentric to the inversed gear and its thickness is identical to the thickness of circular ring 5 μm. The central angle of arc is termed as θ, which is from 90 degree to 170 degree. The simulation results have been 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 420 μm × 420 μm square area. Since the PIT effect occurs in between the super-radiant and sub-radiant resonators, the interaction of the two circular cavities in between the adjacent unit-cell are neglected. The Floquet ports along the z direction are set 3 mm away from the front-side and back-side of MM respectively. Therefore, the simulation can be executed under far-field condition, which is available for plane-wave approximation. The excitation source is a time-domain THz pulse signal from a commercial THz-time domain spectrometer (Menlosystem Tera K15), of which its temporal window is 17 ps. Correspondingly, the temporal interval between two points is 1/3 ps. The transmittance T(ν) as well as the phase spectrum of the MM can be calculated by the function of S-parameter T(ν) = |S21|2. The mesh cell of simulation is 297600. The minimum mesh step is 2.5 and the maximum mesh step is 9.375.
Fig. 1 (a) Schematic diagram of one unit cell of proposed binary MM, in which p = 420 μm, r = 90 μm, w = 4 μm, h = 60 μm, g = 15 μm, respectively. l is the length of metallic arc. KTHz refers to the wavevector of incident THz pulse. ETHz and HTHz refer to the electrical components and magnetic components respectively.
3. Results and discussion
The simulated THz transmittance as a function of frequency versus different central angles of metallic arc is presented in Fig. 2(a) correspondingly. Here, the frequency window is from 0.1 to 0.4 THz since the high-order multipolar resonances are ignored in our works. At the θ = 90 degree, there are two separated transmission minimum occurs on the spectrum initially. One mode is at 0.18 THz termed as νL, the other mode is at 0.36 THz termed as νH. A relatively flat band exists in between the νL and νH. With the θ increases up to 130 degree, an obvious redshift of νH occurs in frequency domain but the νL seems to be isolated to the change of θ. Meanwhile, the aforementioned flat band gradually becomes a transparency maximum νT in THz spectrum. Such a transparency window νT occurs redshift with the νL and νH when the θ increase continuously, but its transmittance become weaker and weaker. When the θ increases up to 170 degree, aforementioned transparency window disappears. The accurate position of resonance modes are listed as below:
Fig. 2 (a) The THz transmittance of MMs with varied central angles from 90 degree to 170 degree at the interval of 20 degree. (b) The 2D THz transmittance as a function of frequency versus different central angles of metallic arc. The step is 2 degree.
In order to get a deep insight into the evolution of transmittance as a function of central angle of metallic arc, the T(ν) is calculated at the step of 2 degree. Thus, one can map the two-dimensional THz spectrums as a function of frequency as well as the central angle, as shown in Fig. 2(b). Obviously, the mode νL locate at 0.18 THz, which is unaffected by the extension of metal arc; while the mode νH occurs monotonically redshift. The mode νL and mode νH unifies at the central angle of 170 degrees. Thus, it is clear that the mode νL attribute to the intrinsic spoof-LSPs resonance on the circular cavity; while the mode νH attributes to the intrinsic mode of metallic arc itself. Since the SPs destructive interference leads to the PIT, the central frequency of transparency window should overlap with the basic resonators. In our case, however, the transparency window is always below the intrinsic frequency of metallic arc but higher than textured circular cavity. At this point, the above transparency window cannot be attributed to the conventionally destructive interference of the intrinsic modes of basic resonators. Actually, 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 above fake transparency window [23].
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 [23,26,27]. Here, the Δτ represent the time delay of THz wave packet instead of the group index. The Δτ can be calculated from the equation as below:
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 our MMs, in which a distinct phase transition is found at the νL and νH modes. The extracted group delays as a function of THz frequency of MMs are illustrated in Fig. 3(b). With the θ increasing from 110 degree to 130 degree, the Δτ increases monotonically from 27.6 ps to 43 ps. To a Lorentz reciprocal binary MM, such a large group delay exceeds the other published records. When the θ increases up to 150, the Δτ achieve 44.6 ps. A finer map of group delay as a function of frequency and θ is simulated in Fig. 3(c), in which the simulated step of θ is 1 degree as well. The resonance side-modes exhibit negative group delay, which is the same as the previous results of PIT effect [27,28]. However, an obviously positive group delay occurs in the transparency windows when the θ is in the range from 120 degree to 168 degree. As such, the slow light is localized in a certain area induced by the interaction between the resonance modes on metallic arc and circular cavity. The maximum on the slow light plateau is at 135 degree, where the group delay achieves 46 ps.Fig. 3 (a) The phase spectra and (b) the group delay of MMs with varied central angles from 90 degree to 170 degree at the interval of 20 degree. (b) The two-dimensional diagram of group delay as a function of frequency and central angle. The step is 2 degree.
Furthermore, another key figure-of-merit for the slow-light device is the product of the group delay time and bandwidth of transparency windows [3]. Here, the Δτ represent the time delay of THz wave packet. The Δν represent the linewidth of transparency window, which is termed as:
Figure 4 shows the product of the group delay time and bandwidth at transparency windows of our MMs with the central angle of metallic arc from 90 to 170 degree at the step of 20 degree. The selected central angles are the same as listed in Table 1. Since our MM is axial symmetry in structure, it is a Lorentz reciprocal system under fundamental time-bandwidth limit [3]. As such, above products are all below 2π.Fig. 4 The product of the group delay time and bandwidth at transparency windows of our MMs with the central angle of metallic arc at 90, 110, 130, 150, and 170 degree, respectively.
Table 1. The mode frequency of basic resonators
In order to reveal the origin and evolution of transparency window νT, the electrical energy density of MM was simulated numerically at the central frequency of νL, νH, and νT correspondingly, which is presented in Fig. 5. Herein, we select the MMs of three different central angles for simulation. The first case is 90 degree, where there is neither transparency window nor slow light behavior. The second case is 135 degree, where the slow light at transparency window achieves maximum value of 46 ps. The third case is 170 degree, where the side-mode νH unified with the side-mode νL. In the first case, a dipolar structure of spoof-LSP dominates the side-mode νL resonance. Therefore, the high-order resonance can be excluded in contributing the transparency window, such as quadrupole, hexapole, octapole, etc. Regarding the unit cell is in uni-axial symmetry, only dipolar resonance is able to be excited due to the charge oscillation in resonators, which can strongly couple with the incident THz wave. The dipolar spoof LSP becomes the intrinsic mode of circular cavity so that the resonance frequency is isolated to the extension of metallic arc since the electrical energy on metallic arc can be neglected at the side-mode νL. Correspondingly, an intrinsic dipolar spoof-LSP is found on the metallic arc dominates the side-mode νH. Meanwhile, another parasite spoof-LSP occurs at the lower-halve of circular cavity, which is much smaller than the spoof-LSP of side-mode νL. Normally, the shorter the dipole oscillator is, the higher the resonance frequency is. It is obvious that the polarity and length of these two spoof-LSPs are the same. Therefore, a hybridation of above two the spoof-LSPs on metallic arc and on circular cavity contribute to the side-mode νH. Compared to the first case, the metallic arc extends its length up to 135 degree so as to result in a redshift of the side-mode νH. Meanwhile, the intrinsic mode νL of circular cavity does not change. At the transparency windows, a new-born dipolar spoof-LSP appears on the circular cavity with opposite polarity to the spoof-LSP on metallic arc. In accordance with the Faraday's law of induction, the dipolar spoof-LSP on metallic arc is an alternative current source, which induces a magnetic flux along the direction of incident THz wave-vector [39]:
where ψ is the magnetic flux, B is the magnetic flux density, x and y is the accurate position in area below metallic arc. Owing to the electromagnetic induction, another induced magnetic flux in the close area of circular cavity is generated opposite to the direction of THz wave-vector. The induced magnetic field interacts with circular cavity to produce an electromotive force. The conservative electrostatic field created by separation of charge exactly cancels the forces producing the electromotive force. Thus, the electromotive force has the same value but opposite sign as the integral of the electric field aligned with an internal path between two terminals of a source of electromotive force in metallic arc. Mathematically [39]:Here, U is the electromotive force; t is varied time; Earc is the electrical field on metallic arc, which induces the electromotive force on metallic arc; l is the length of metallic arc; s is the spatial distance. Thus, a couple of opposing dipolar spoof-LSP dominates the transparency windows. Therefore, the two spoof-LSPs of opposite direction lead to a fake quadrupole, which will repel each other in magnetic dipole momentum, as is verified in another PIT behavior [40]. As such, the slow light achieves maximum with the induced spoof-LSP is the same as the origin spoof-LSP on metallic arc in oscillation strength. In the third case, the intrinsic spoof-LSP on circular cavity is coupled in-phase with intrinsic spoof-LSP on metallic arc when the central angle θ increases up to 170 degree. These two modes oscillate in phase so that the mode νL unifies with the mode νH. Thus, the transparency window closes.Fig. 5 Electrical field distribution of side-modes and THz transparency window of MMs at 90 degree, 135 degree, 170 degree, correspondingly. The νT, νL and νH refer to the transparent window, low-frequency side-modes and high frequency side-modes respectively. The color bar refers to the polarity as well as the relative strength of electrical field. Crosshatching means no data.
4. Summary
In summary, a maximum of 46 ps group delay of THz slow light phenomenon owing to the spoof-LSP transparency is predicted via numerical simulation. A binary meta-atom composed of a metallic arc and a textured closed cavity of periodically grooves support the spoof localized surface plasmon (LSP). By tuning the central angle θ of the arc from 90 degree to 170 degree, the transparency window appears to be an open-and-close behavior. A localized THz slow light is presented in the transparency window. The numerical mapping of electromagnetic field indicates that a destructive inference occurs in between the excited and induced spoof LSP dipole resonances, which results in the transparency window. The group delay achieves maximum at the θ of 135 degree.
Funding
Joint Research Fund in Astronomy (Grant No. U1631112) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS).
Acknowledgments
Zhenyu Zhao and Yana Chen contribute equally in this work.
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