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Abstract
We experimentally investigate the terahertz (THz) electromagnetically-induced transparency (EIT)-like phenomenon in a metamolecule (MM) of three-body system. This system involves a couple of geometrically identical split-ring resonators (SRRs) in orthogonal layout conductively coupled by a cut-wire resonator. Such a three-body system exhibits two frequency response properties upon to the polarization of incident THz beam: One is the dark-bright-bright layout to the horizontally polarized THz beam, where there is no EIT-like effect; the other is bright-dark-dark layout to the vertically polarized THz beam, where an EIT-like effect is observable. The transparency window can be tuned from 0.71 THz to 0.74 THz by the displacement of cut-wire inside the trimer MM. A maximum of 7.5 ps group delay of THz wave is found at the transparent window of 0.74 THz. When the cut-wire moved to the mid-point of lateral-side of SRR, the EIT-like phenomenon disappears, this leads to a localized THz slow-light effect. The distribution of surface currents and electric energy reveals that the excited inductive-capacitive (LC) oscillation of bright-SRR dominates the high frequency side-mode, which is isolated to the displacement of cut-wire resonator. However, the low frequency side-mode originates from the constructive hybridization of LC resonance in dark-SRR coupled with a localized S-shaped dipole oscillator, which is tunable by the displacement of cut-wire. As a consequence, the group delay as well as the spectral configuration of transparency window can be manipulated by tuning one side-mode while fixing the other. Such an experimental finding reveal the EIT-like effect in a conductively coupled three-body system and manifests a novel approach to achieve tunable THz slow-light device.
© 2017 Optical Society of America
1. Introduction
A destructive interference of surface plasmons (SPs) inside the artificial meta-molecule (MM) will give arise to a transparency window over a broad absorption spectrum from optical frequency to terahertz band, which can mimic the electromagnetically induced transparency (EIT) phenomenon of atomic systems [1–13]. The group velocity of light pulse passing through this transparency window will be reduced grammatically owing to the giant group dispersion. Therefore, the MMs exhibiting EIT-like effect becomes a potential device to greatly reduce noise in telecommunication so as to allow information to be transmitted more efficiently, which attracts much attention. Any MM made of binary resonators such as split-ring resonators (SRR) [4–6], cut-wires [7,8], and U-shape resonators [9,10], can support the EIT-like phenomenon as long as the resonance frequencies of aforementioned two coupled resonators are nearly identical with small deviations, but the quality factors are in strong contrasting. The spectral configuration of EIT-like effect and slow light can be manipulated by controlling the geometric layout of basic resonators inside MM [14–19], or by utilizing an external stimulus, such as optical pump, temperature, magnetic fields, and bias voltage [20–25]. Aforementioned EIT-like phenomena all occur in the dimer MM based on bright-and-dark resonators or superradiant-and-subradiant resonators. Therefore, the established EIT-like phenomena naturally are all based on the dimer MM, which is a two-body problem in physics. If one more SP resonator is introduced into the dimer MM, the destructive interference of SPs is able to be affected by the interaction of the third coupled SP resonator. In nonlinear optics, multiple photons can induce coherence effects in a dissipative quantum system, which makes it achievable to control photon by photon, such as four-wave mixing and quantum coherent control techniques [26–28]. In agreement to the principle of optics, the coupled systems feature phenomenon ascribed to a coherent control of transitions in a three-level atomic system. As such, it inspires one to control the multiple SPs interaction. Furthermore, the recent work indicates that the SP can be manipulated in one direction via asymmetric design of MM [29], which paves a way to control the two SP interactions using the third one. Interestingly, the THz response of conductively coupled trimer MM exhibit Möbius optical symmetry [30], which is unachievable in a MM of two-body system. Thus, a conductively coupled three-body MM system maybe possesses different EIT-like behavior compared to a two-body MM system. However, it is not realized yet.
In this work, we investigate the THz response of the trimer MM made of conductively coupled dark -bright-bright resonators and bright-dark-dark resonators, respectively. Since the strength of conductive coupling exceeds that of near-field coupling [31,32], a relatively larger group delay has been successfully achieved in the transparency window of EIT-like effect [33]. Therefore, we design MM using three connected resonators. The THz response of the trimer MM is calibrated using THz time-domain spectroscopy (THz-TDS). It is found that the EIT-like-phenomenon only occurs in the trimer MM of bright-dark-dark resonators layout. Then, the EIT-like spectral configuration and slow light is tuned by the displacement of contact point of the three resonators. With the help of surface current analysis and electromagnetic field simulation, the mechanism of EIT-like effect in the MM of three-body system is discussed.
2. Experiment
The structural elements of a trimer MM is shown in Fig. 1(a), which is made of a dimer MM (two rectangular SRRs) and a monomer MM (cut-wire) connected conductively. The lateral-side of SRR is 24 μm in length. The gap is 10μm in width. These two SRRs are separated 66 μm one another in horizontal direction. The cut-wire is 66 μm in length, as is identical to the displacement between the two SRRs, and its width is 4 μm, the same as SRRs. The Fig. 1(b) shows a resonance mode at 0.8 THz appears in the separated when both are irradiated normally by the incident THz pulse in horizontal and vertical polarization. This resonance frequency is identical to the individual cut-wire when the polarization of incident THz pulse is along the X-axis. However, the cut-wire has no response to the incident THz of vertical polarization. When the THz polarization is parallel to the gap of SRR, a circular current loop indicate that the inductive-capacitive (LC) resonance dominate the resonance modes. Herein, one SRR plays the role as a bright resonator and the other as a dark resonator with respect to the corresponding polarization of incident THz pulse. Therefore, the two separated rectangular SRRs are termed as dimer MM. The uni-directional currents indicate that the dipole oscillation dominate the resonance mode of cut-wire. Therefore, it plays the role as bright resonator to the horizontally polarized THz pulse, but as a dark resonator to the incident THz in vertical polarization. Here, it is termed as monomer MM. Here, the monomer MM connected the separated two SRRs of the dimer MM, which results in a trimer MM. The method of sample fabrication and characterization is the same as our previous works [34–38].The resonators patterns of MM are fabricated on a piece of 625 μm-thick semi-insulating gallium arsenide (SI-GaAs) substrates by photolithography. A metal layer of 120 nm thick gold (Au) and 5 nm thick titanium (Ti) is deposited on the patterned substrate. The Ti acts as an adhesion layer between Au and SI-GaAs. The unit cell of each MM is in the rectangular area of 140 μm × 70 μm. The sample size of MM is 1 cm × 1 cm. The transmission spectra of the samples were measured by a conventional THz-TDS system. A Ti: Sapphire oscillator (Mai-Tai, Spectra-Physics GmbH) is used for ultrafast optical excitation and a pair of low-temperature grown GaAs photoconductive antennas was used as THz radiation emitter and sensor. The THz emission was collimated onto the metal layer of MM by a couple of off-axis parabolic mirrors (OAPM) with a diameter of 50.8 mm. The transmitted THz wave was collimated by another couple of OAPM onto the sensor. The signal was read out into a Lock-In amplifier (SRS 272, Stanford Instruments) at the time constant of 100 ms. The whole measurement was carried out in dry nitrogen environment to avoid absorption of water vapor. The resonance modes are recorded in the frequency range from 0.4 THz to 1.2 THz. A bare SI-GaAs wafer identical to the sample substrate served as a reference. The transmission spectrum is extracted from Fourier transforms of the measured time-domain electric fields, which is defined as [34–38]:
where Esample(ν) and Eref(ν) are the Fourier transformed electric fields through the sample and reference, respectively. T(ν) is the transmittance as a function of THz frequency. Such a trimer MM works in bright-dark-dark layout to the vertically polarized THz pulse, but in dark-bright-bright layout to the horizontally polarized THz pulse. Finally, a finite difference time domain (FDTD) algorithm based software CST Microwave StudioTM was used to simulate the THz transmittance of samples as well as the electromagnetic field at resonance modes correspondingly.
Fig. 1 (a) Microscopic images of the basic resonator of monomer, dimer, and trimer MM, in which L = 140 μm, h = 70 μm, l = 24 μm, g = 10 μm, w = 3 μm, d = 66 μm, respectively. (b) The THz response of separated SRRs and cut-wire under differently polarized THz incidence, b1) and b2): vertical polarization, b3) and b4): horizontal polarization. Insets: The THz-induced surface currents of separated SRRs and cut-wire under vertically and horizentally polarized THz incidence correspondingly.
3. Results and discussion
Then, an introduction of relative displacement of conductive junction point from the top to the bottom of SRR leads to a displacement δ with the step of l/4 length of the lateral-side of SRR. The schematic diagram of the conductively coupled trimer MM is presented in Fig. 2(a). The number I, II, III, IV, and V refer to the MM of different value of displacement δ from top to bottom. Correspondingly, the measured and simulated transmittances of the trimer MM under bright-dark-dark layout and dark-bright-bright layout are illustrated in Figs. 2(b) and 2(c), correspondingly. Here, we address that THz scattering owing to the inevitable imperfection at edge of each unit cells, which results in slight deviations between the measurement and the simulation [34,35]. To the bright-dark-dark layout of the trimer MM (the polarization of incident THz pulse is parallel to the Y-axis), a transparency window occurs in the THz frequency spectrum between the two resonances side-modes initially. Herein, we define the frequency of transparency windows as νT and the low frequency side-modes as νL, while the high frequency side-modes as νH, respectively. With the increase of displacement δ from 0 to l/2, the transparency window closes gradually. The central frequency of transparency window is at around 0.71 THz.When the cut-wire moves to the mid-point of lateral side of SRR, the transparency windows νT completely closes. However, such a window open again with the displacement δ increasing from l/2 to l. The central frequency of transparency window shifts from 0.73 THz to 0.74 THz. The depth variation of transparency windows νT appears to be in V-shape. Interestingly, the low-frequency side-mode νL shows a V-shape variation as well. However, the strength and frequency variation of high frequency side-mode νH, almost can be neglected. Such a phenomenon is completely different from reported EIT-like effect in other MM of two-body system [1–13]. To the dark-bright-bright layout of the trimer MM, however, a single resonance mode νS at 0.35 THz is activated when the polarization of incident THz beam is parallel to the X-axis. The spectral configuration of this single mode seems to be isolated to the displacement of cut-wire even though the δ increases from 0 to l. As a consequence, the EIT-like phenomenon only takes effect to the bright-dark-dark layout of the trimer MM but no effect to the dark-bright-bright layout. Therefore, the following analysis and discussion only focus on the EIT-like phenomenon under the bright-dark-dark layout of the trimer MM.
Fig. 2 (a) Schematic diagram of trimer MM (b) THz transmittance of trimer MM in bright-dark-dark layout to vertically polarized THz pulse. Here, the νT refers to the central frequency of transparency windows; The νL and νH refers to the the low frequency side-mode and the high frequency side-mode, respectively. (c) THz transmittance of trimer MM in dark-bright-bright layout to horizontally polarized THz pulse. The νS refers to the central frequency of single resonance mode. Blue solid-line refers to the simulated THz transmittance. Red solid-line refers to the measured THz transmittance.
A much finer simulation of the THz transmittance of MM as a function of δ and frequency are illustrated in Fig. 3, in which the simulated step of displacment is 1 μm. In agreement with the experimental data shown in Fig. 2(b), the high frequency side-mode νH maintains the resonance strength over the range of transmission frequency spectrum with the δ increasing from 0 to l. Besides, the frequency shift is almost negligible. To the low frequency side-mode νL, its central frequency changes subtly. However, its oscillation strength decreases in the beginning while increases in the end, which exhibits V-shape. Such a phenomenon is totally different from the other MMs of two-body system in previous works [1–25].
Fig. 3 The 2-dimensional map of THz transmittance as a function of THz frequency and the displacement value δ at the step of 1 μm.
A significant evidence of EIT-like effect is a positive group delay (Δtg) at the transparency window in spectrum [5,20]. Here, the Δtg represent the time delay of THz wave packet instead of the group index. The Δtg can be calculated from the equation as below [5,20,33]:
where φ and ν refer to the effective phase and frequency of THz complex transmission spectrum, respectively. To determine the φ, the phase of incident THz wave is subtracted, traveling in the free-space between input port and the metal layer of MM, from the phase between the input and output port. As such, only the desired phase difference between free-space and the output port which is positioned 625 μm behind the MM. From the measured spectra, however, the phase of free-space is initially subtracted from the measured phase of MM. An additional phase delay of free-space with the thickness of 625 μm was manually added to the subtraction. The effective phase can be calculated from the equation as below [5,20,33]:here, φT is the measured phase spectrum of our MM, and φref is the phase spectrum of reference; k is the wave-number of free space and D is the distance between input and output ports.Figure 4(a) shows the measured phase spectrum of our MM, in which a distinct phase transition is found at the νL and νH modes. Following aforementioned retrieval method of group delay, an obvious group delay occurs at the frequencies of transparency windows in MM, which is shown in Fig. 4(b). With the δ increasing from 0 to l/4, the Δtg decreases monotonically from 6.6 ps to 5.2 ps. However, a further increase of δ reduces the Δtg from 3.5 ps to 7.5 ps with the δ increasing from 3l/4 to l. Such a V-shape variation of group delay indicates that the dispersion reaches maximum at a certain area where the cut-wire contacts the corner of SRRs, and slow light is localized as well. A finer map of group delay as a function of frequency and δ is simulated in Fig. 4(c), in which the simulated step of δ is 1 μm. The resonance side-modes exhibit negative group delay, which is the same as reported EIT-like effect in dimer MM [33–37]. However, an obviously positive group delay occurs in the transparency windows when δ = 0 and δ = l. Since a EIT-like effect of MM originates from the destructive interference of two surface plasmons (SPs), the central frequency of transparency window should overlap with the basic resonators. In our case, however, the transparency window is always below the frequency of SRRs and cut-wire resonators. Furthermore, the SP propagating at the metal-GaAs interface is naturally an electron density wave, which is driven by the electric field of incident THz wave; thus, the propagating direction of SPs must be along the wave polarization at normal incidence. To the trimer MM, however, only the LC oscillation of left-SRR can be excited by the incident THz pulse. The right-SRR and the cut-wire are dark resonators are unable to support horizontal oscillations of SPs excited by the vertically polarized THz pulse. At this point, the above localized slow light is not owing to the destructive interference of the intrinsic modes of bright and dark resonators. A further analysis of surface currents will help to reveal the origin of aforementioned EIT-like phenomenon.
Fig. 4 (a) The measured phase spectra and (b) the group delay of trimer MM. I, II, III, IV and V, refer to the displacement δ of 0, l/4, l/2, 3l/4, l, correspondingly. Here, the νT refers to the central frequency of transparency windows. (c): The 2-dimensional map of THz group delay as a function of δ and THz frequency.
The surface currents of νL and νH sidemode are shown respectively in Fig. 5. Obviously, the νL sidemode is composed of two localized currents. One current flows from the upper-lateral side of the left-SRR to the left-half of cut-wire via bottom-line of the left-SRR, which forms an S-shape loop; the other one is a circulating loop localized in the right-SRR. With the displacement δ increasing from 0 to l/2, the S-shape current and circulating current disappears gradually. A further increasing of δ from l/2 to l change the circulating currents loop from clockwise to counter-clockwise. Simultaneously, the flowing direction of S-shape current is reversed as well. On the contrary, a clockwise circulating current loop on the left-SRR dominates the resonance of νH sidemode, and the displacement of cut-wire has no influence on the current direction and strength of νH sidemode. Since the circulating current is the evidence of the LC resonance [39], we proposed that the νL sidemode is attributed to the constructive hybridization of LC resonance and S-shape dipole oscillator, while the νH sidemode is naturally the bright mode of left-SRR. The displacement of cut-wire decouple the S-shape oscillator and the LC resonance in right-SRR, which terminates the νL sidemode at δ = l/2. To the νH sidemode, however, the other two dark resonators (cut-wire and right-SRR) are not excited. Only the bright mode left-SRR contributes to the resonance. At this point, the transparency window is constructed by the νL and νH modes rather than the expected destructive interference between intrinsic bright and dark resonators. Actually, the displacements of conductive contact points give rise to the phase flip-flop of a constructively coupled resonance composed of S-shape dipole and LC oscillator.
Fig. 5 Surface currents of the trimer MM at the mode of νL and of νH. Color bars: The relative strength of currents and magnetic energy. I, II, III, IV and V, refer to the displacement δ of 0, l/4, l/2, 3l/4, l, correspondingly.
Aforementioned arguments can be supported by calculating the complex permittivity as a function of THz frequency as below [40]:
The permittivity can be derived from the simulated parameters of S11 and S21 calculated by CST Microwave StudioTM software. Initially, one can achieve the effective refractive index n and impedance z following the equation below [40]: Here, the permittivity ε is directly calculated from ε = n/z.Figure 6 shows the retrieved complex permittivity of the MM with different deviation respectively. Here, we address that above retrieval is based on the simulated S-parameters. To the modes of νL and νH, the real part of the function of complex permittivity εr shows a large negative value, while the imaginary part εi shows large positive values, describing a lossy medium at these frequencies. Since the dielectric function is flat the frequency of νT modes, which is the same as other THz EIT-like effect in the dimer MM. Comparing the oscillation strength of εr at the frequency of νL and νH shown in Fig. 6, it is evident that the real part and imaginary part of permittivity of νL exceed that of νH at δ = 0 and δ = l. A further displacement of δ reduces the permittivity, which is in agreement with the variation of oscillation strength of νL mode and νH mode as well. In agreement with the basic principle of dielectrics of solid-state material, the permittivity can be written as a function of polarization as below [41]:
where, ν is the frequency, and the χ is the a tensor of polarizability. From the observable far-field quantities, the properties of the SP dipole oscillators can be derived to govern the response of the three wires (2 lateral-sides and 1 bottom-line) forming the SRR. Correspondingly, its polarizability can be written as below [42]:here, χxx and χyy are susceptibility tensors for the two polarization direction (x and y), which is diagonal. The eigenmodes of effectively homogenous medium are orthogonal so that the SRR is bianisotropic in dielectric function, which leads to a negative εr and positive εi occurs at LC oscillation frequency. To the S-shape dipole oscillator, however, its polarizability is more complex [42,43]:Fig. 6 The retrieved frequency-dependent dielectric functions of MM. The parameters are from the simulated S-parameters. The red lines refer to the real permittivity εr. The blue curves refer to the imaginary permittivity εi. I, II, III, IV and V, refer to the displacement δ of 0, l/4, l/2, 3l/4, l, correspondingly.
Obviously, the susceptibility tensor of S-shape oscillator is off-diagonal. According to the Onsager-Casimir principle [44–46], the tensor is in symmetry along the diagonal elements owing to the time reversal so that the χxy must be identical to the χyx in Eq. (10). Therefore, its dielectric function is in elliptical dichroism. To the S-shape oscillator, the εi occurs negative at the resonance frequency if the off-diagonal elements are not zero [42]. In our case, however, εi always shows positive at resonance frequency in dielectric function all over the THz range. Therefore, we propose that the off-diagonal elements of susceptibility of S-shape dipole oscillators are zero. Thus, a constructive hybridization in between the Lorentz resonance of S-shape dipole oscillator and LC resonance of SRR dominates the νL mode. Meanwhile, the permittivity at the νH mode does not change obviously even though the δ shifts from 0 to l. In agreement to the circulating current shown in Fig. 5, it originates from an intrinsic LC resonance in the left-SRR of the trimer MM. Therefore, the transparency window is constructed by the two independent side-modes: νL and νH mode. The displacement of cut-wire only has influence on the strength modulation of νL mode. Regarding the constructive hybridization of S-shape oscillator and LC resonator results in the νL mode, the geometry of the MM plays a key role on this hybridization. The bright-SRR and dark-SRR is in orthogonal layout in each MM so that the diagonal tensors of two SRRs are in orthogonal as well. Here, we propose that the susceptibility of bright-SRR counteract with the dark-SRR when the cut-wire shift to the middle-line between the two SRRs. Then, the νL mode disappears. Since the transparency window νT is constructed by the νL mode and νH mode, the νT disappears as well.
The evolution of νL and νH mode can be extracted from mapping the spatial distribution of electromagnetic field of the trimer MM [47–50].
Figure 7 shows the simulated THz magnetic field strengths of MM along the incident THz wave-vector, respectively. In accordance with the Ampère's right hand screw rules, the direction of circulating surface current at νH modes is counter-clockwise in the left half of MM, which induces a magnetic flux out-of-plane. As shown in Fig. 7, the magnetic flux opposite to the direction of incident THz wave-vector is larger than that along the direction of incident THz wave-vector. It indicates that the net magnetic dipole of νH modes is positive in the trimer MM. To the νL modes, however, it is attributed to the two localized magnetic dipoles shown in Fig. 7. In the left area to the black dash-line, the S-shape dipole oscillator forms one magnetic dipole. Its south pole is restricted in the closed area of dark-SRR, and its north pole is in the quarter turn between the bottom-line of SRR and the cut-wire. The polarity of this magnetic dipole is not changed by the displacement of cut-wire. To the right area of black dash-line, the clockwise current on dark-SRR forms another magnetic dipole. However, its polarity will be reversed when the cut-wire moves beyond l/2. Such a flip-flop of polarity of localized magnetic dipole changes the net magnetic dipole of νL mode, which originates from the constructive hybridization of LC resonance and S-shape dipole oscillators. Thus, the low-frequency side-mode can be tuned by the displacement of cut-wire. As a consequence, the group delay as well as the spectral configuration can be manipulated by tuning one side-mode while fixing the other. In practice, our experimental findings manifest a novel approach to tune the slow-light at THz frequency range, and extend the EIT-like phenomenon from two-body problem to three-body problem.
Fig. 7 Magnetic field distributions of trimer MM at the mode of νL and of νH. The red color and blue color refer to the strength and the direction of magnetic fields along ( + : north pole) or opposite (-: south pole) to the direction of incident THz wave-vector, correspondingly.
4. Summary
In summary, a localized terahertz (THz) electromagnetically-induced transparency (EIT)-like phenomenon inside trimer meta-molecules (MM) is investigated experimentally. This MM involves a couple of geometrically identical split-ring resonators (SRRs) in orthogonal layout conductively coupled by a cut-wire resonator. Each basic resonator manifests the same resonant frequencies when being excited by the incident THz radiation. When the polarization of incident THz beam is parallel to the cut-wire, the trimer MM works in dark-bright-bright layout and there is only single resonance occurs in frequency spectrum. When the polarization of incident THz beam is perpendicular to the cut-wire, the trimer MM works in bright-dark-dark layout in which an EIT-like phenomenon occurs. The transparency window can be tuned by the displacement of cut-wire inside the MM. A maximum of 7.5 ps group delay of THz wave is found at the transparent windows of 0.74 THz. When the cut-wire moved to the mid-point of lateral-side of SRR, the EIT-like effect disappears, and the THz slow-light effect is localized. The distribution of surface currents and electric energy reveals that the excited inductive-capacitive (LC) oscillation of bright-SRR dominates the high frequency side-mode, which is isolated to the displacement of cut-wire resonator. However, the low frequency side-mode originates from the constructive hybridization between a dark-SRR and a localized S-shaped dipole oscillator, which is tunable by the displacement of cut-wire. As a consequence, the group delay as well as the spectral configuration of EIT-like effect can be manipulated by tuning one side-mode while fixing the other. Our findings reveal the EIT-like effect in a MM of three-body system.
Funding
This work is financially supported by the National Natural Science Foundation of China (NSFC) (61307130) and the Joint Research Fund in Astronomy (U1631112) under cooperative agreement between the NSFC and Chinese Academy of Sciences (CAS). Z.Z. acknowledges the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry as well as Innovation Program of Shanghai Municipal Education Commission (14YZ077). W.P. acknowledges the Strategic Priority Research Program (B) of the CAS (XDB04030000).
Acknowledgments
Zhenyu Zhao and Xiaobo Zheng contribute equally in this work.
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