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Abstract
We demonstrate a localized slow light phenomenon in a symmetry broken metamolecule (MM) of conductively coupled dark resonators at a terahertz band. Under a dark-mode excitation condition, the single mode resonance becomes dual modes by breaking the uniaxial symmetry of MM. Thus, a transparency window exists in between dual modes. An interaction of V-shaped plasmonic antenna-type (VA) resonances results in a plasmon-induced transparency (PIT) when the asymmetric deviation is below 13 μm. A maximum 25.9 ps group-delay of incident THz pulse is observed at the transparency window. When the asymmetric deviation is beyond 13 μm, one excitation pathway switches from VA resonance to the inductor-capacitor (LC) resonance, which dominates the high-frequency side-mode. Then, the PIT effect transfers to the PIT-like behavior and the slow light phenomenon vanishes. The aforementioned discovery allows for a speed modulation of slow light via symmetry breaking in MM.
© 2017 Optical Society of America
1. Introduction
Electromagnetically induced transparency (EIT), a quantum effect originates from the destructive interference of the transition probability amplitude between atomic states excited by incident electromagnetic fields, which creates an extreme large dispersion within a narrow transparency window over a broad absorption spectrum, and it reduces the group velocity of propagating light dramatically [1–3]. Such a slow light effect could reduce the noise obviously so as to improve the information transmission efficiency in telecommunication [4,5]. Furthermore, the slow light effect could cut power requirements even up to million-fold of optical switching devices [6]. Owing to the energy interval between the quantum states for transitions in atoms, the EIT-based slow light effect is mostly limited in optical frequency range. However, the rapid development of THz telecommunication demands effective signal buffers and noise filters. As such, the THz slow light devices become a potential key component in THz telecommunication. Alternatively, plasmon induced transparency (PIT) effect - an analogue of EIT effect provides a cost-effective approach to achieve slow light from optical frequency to THz band by using artificial meta-molecules (MM) [7–11]. The MM are composed of multiple electromagnetic resonance elements, such as split-ring resonators (SRR) [12], cut-wires [13], and U-shape resonators [14,15]. These basic resonators support a destructive interference of surface plasmons (SPs), which create a slow light channel at the transparency window over the frequency spectrum. The spectral profiles of PIT, such as the width of transparency window and the group delay, can be manipulated by tuning the coupling strength in between either the bright-and-dark resonators, or the superradiant-and-subradiant resonators of MM. The bright resonators, including the superradiant-and-subradiant resonators, can be directly excited by the incident radiation, while the dark resonator is unable to be excited directly by the incident radiation, but rather by the bright resonator via near-field coupling. Up to date, the maximum group delay of THz pulse owing to the PIT-effect is limited below 10 ps [16,17]. Compared to the bright resonators, the dark-resonators exhibit higher Q factor [18–21]. Herein, it may be available to achieve a larger dispersion channel by manipulating the destructive inference of surface plasmon in between the different dark resonators. Although the dark-resonator is unable to be excited directly by the incident THz pulse, an extreme high Q factor can be achieved owing to the conductive coupling in between dark-and-dark resonator of MM [22,23]. It is well-known that the conductive coupling has higher efficient than the near-field coupling [24]. Hence, a MM of conductively coupled dark-and-dark resonators maybe become a new approach to achieve a slower light at THz frequency band, but not realized yet.
In this work, we demonstrate a localized THz slow light phenomenon in a series of symmetry broken MM of conductively coupled dark resonators. A maximum 25.9 ps group delay is observed at an approximate asymmetric deviation. Correspondingly, spectral evolution of MM is recorded by the THz-time domain spectroscopy. With the help of numerical simulation of surface currents as well as electromagnetic field distribution, the mechanism of localized slow light effect is discussed.
2. Experiment
The structural elements of a triangle SRR (TSRR) is shown in Fig. 1(a), which composed of two lateral-side and a bottom-line including two arms and a gap in the middle. The lateral-side of TSRR is 25.5 μm in length. The gap is 4μm in width. The arm is 16 μm in length. The apex-angle of TSRR is π/2 radian. Unlike the rectangular SRR or circular SRR, the parasite dipole oscillation is minimized in TSRR, but the inductive-capacitor (LC) resonance mode is maintained [25]. The cut-wire is of 54 μm in length and 4 μm in width. The Fig. 1 (b) shows a resonance mode at 0.8 THz appears in both single TSRR and individual cut-wire when the polarization of incident THz pulse is along the X-axis (horizontal polarization). Compared to the TSRR, the resonance mode of cut-wire exhibits a relatively low Q factor. However, both resonators exhibit no response in THz spectrum when the polarization of incident THz pulse is along the Y-axis (vertical polarization). Herein, both TSRR and cut-wire play the role as dark resonators for an incidence of vertically polarized THz wave. Then, the TSRR is conductively coupled in the mid-point of the cut-wire so that the MM is in uni-axial symmetry (along the Y-axis). Then, an introduction of relative displacement of conductive junction point from the initial symmetric axis leads to an asymmetric deviation δ, which increases from 0 to 25 μm at the step of 5 μm. The schematic diagram of symmetric broken MM is presented in Fig. 1 (c). The process of sample fabrication and characterization is the same as our previous works [26,27]. 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 80 μm × 80 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, as shown in Fig. 1(d). A Ti: Sapphire oscillator (Mai-Tai, Spectra-Physics) 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 CSRR 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 0.2 THz to 1.5 THz. A bare SI-GaAs wafer identical to the sample substrate served as a reference. The THz radiation is in normal incidence onto the metal layer of MM. The transmission spectrum is extracted from Fourier transforms of the measured time-domain electric fields. 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. 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 the elements of MM. (b): The THz transmittance of TSRR and cut-wire under bright and dark excitation condition, and the polarization of incident THz wave. (c): Illustration of the symmetry broken MM. (b): The diagram of THz transmittance measurement.
3. Results and discussion
The measured and simulated transmittances of MM samples are illustrated in Fig. 2(a). Initially, the cut-wire connects with the TSRR in the mid-point of bottom-line along the X-axis so that the MM is in uni-axial symmetry. A dark mode at 0.8 THz termed as νH is activated when the polarization of incident THz beam is parallel to the Y-axis. Then, the TSRR is displaced away from the symmetric axis; an asymmetric deviation δ is induced in our MM from number II to VII. A second dark mode appears in the THz spectrum below the first dark mode. Meanwhile, a transparency window in between the two dark modes is constructed. Herein, we define that the low frequency side mode as νL, while the high frequency side mode as νH, respectively. Correspondingly, the linewidth of νL and νH are termed as ΔνL and ΔνH. The parameters of symmetry breaking induced broadband PIT effect are listed in Table 1.
Fig. 2 (a): THz transmission of the symmetry broken MM. Blue solid-line: experimental measurement, red solid-line: numerical simulation. (b): The 2-dimensional map of THz transmittance as a function of asymmetric deviation and THz frequency. (c): The frequency-dependent dielectric functions of samples. Purple solid-line: real permittivity εr. Green solid-line: imaginary permittivity εi.
Table 1. Resonance properties of side-modes
Here, we address that the oscillation strength of the second dark mode TL increases monotonically, while that of the first dark mode TH decreases monotonically. This phenomenon is very much like a teeter-totter effects in complementary SRR-based MM [24]. 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 [24,25]. A much finer simulation of the THz transmittance of MM as a function of δ and frequency are illustrated in Fig. 2 (b), in which the simulated step of δ is 1 μm. In agreement with the experimental data shown in Fig. 2 (a), both modes appear redshift over the range of transmission frequency spectrum with the δ increasing from 0 μm to 13 μm. To a further increasing of deviation (13 μm < δ < 25 μm), the νL exhibits redshift behavior, but the central frequency of νH almost does not change. Aforementioned variation trends of oscillation strength of side-modes νL and νH can be extracted by the complex permittivity as a function of THz frequency as below [28]:
The real part εr and image part εi of permittivity can be derived from the measured data, as is shown in Fig. 2 (c). The real part of the function of complex permittivity shows a large negative value, while the imaginary part shows large positive values, describing a lossy medium at these frequencies. When the δ of MM increases from 0 μm to 5 μm, a relatively weaker permittivity suddenly appears below the νH mode in the dielectric spectrum, which is in agreement with the frequency position of νL. A further enlargement of δ results in a monotonic enhancement of νL as well as a reduction of νH mode, accompanied with the redshift in dielectric spectrum, which is in good agreement with the evolution of THz transmittance of side-modes shown in Table 1. Simultaneously, the permittivity functions become flat at the central frequencies of transparency windows. Although the destructive interference of SPs (PIT-effect) could leads to a flat permittivity at the transparency window [17], a frequency detuning in between the two resonators (PIT-like behavior) exhibit similar flat permittivity as well [29]. Therefore, one needs a further analysis to discriminate aforementioned mechanisms.A significant evidence of PIT effect is a positive group delay (Δtg) at the transparency window in spectrum. 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 [16,17]:
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 [16,17]: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 3(a) shows the measured phase spectrum of our MM, 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 MM are illustrated in Fig. 3(b). With the asymmetric deviation δ increasing from 5 μm to 10 μm, the Δtg increases monotonically from 19.1 ps to 25.9 ps. Such a large group delay exceeds the other published records [16,17]. When the δ increases up to 15 μm, the Δtg decreases monotonically from 25.9 ps to 0 ps. However, the Δtg become invisible with the δ increasing from 15 μm to 25 μm. Obviously, the observed slow light is localized at a certain range of asymmetric deviation. A finer map of group delay as a function of frequency and δ is simulated in Fig. 3 (c), in which the simulated step of δ is 1 μm. The resonance side-modes exhibits negative group delay, which is the same as other reported PIT effect [16,17]. However, an obviously positive group delay occurs in the transparency windows when the δ is in the range from 2 μm to 13 μm. The simulated maximum group delay achieves 27 ps when the δ reaches 8 μm. Since a PIT-effect of MM originates from the destructive interference of 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 TSRR 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. In our case, however, both the TSRR and the cut-wire are dark resonators, which are unable to support horizontal oscillations of SP 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 dark resonators shown in Fig. 1(b).Fig. 3 (a): The experimental measured THz phase spectra of the symmetry broken MM. (b): The experimental measured THz group delay spectra of the symmetry broken MM. (c): The 2-dimensional map of THz group delay as a function of asymmetric deviation δ and THz frequency.
Our recent works indicate that the direction of surface currents flows of side-modes is in mirror symmetry [17]. To the PIT-like behavior, however, the surface currents of side-modes are totally different [29]. Herein, the origin of above localized slow light phenomenon can be revealed via the numerical analysis of THz-induced surface currents as well as the magnetic field distribution of MM, which are shown in Fig. 3. To the MM of number I, there is no circulating current in the TSRR so that the LC resonance is excluded to the generation of νH mode. Alternatively, a couple of opposite currents flows on the cut-wire while the current flows along the lateral-sides of TSRR are in mirror symmetry. The opposite current means a destructive coherence, which will counteract the oscillation, while the currents on lateral-sides make the TSRR much more like a symmetric V-shaped plasmonic antenna (VA), which consisting of two orthogonal resonating elements [30,31]. Correspondingly, its first-order symmetric mode of VA can be extracted from the formula as below [32–34]:
Here, l is the length of arm of the equivalent VA (l = 25.5 μm), nres is the effective refractive index at resonance frequency (nres = 7.4), which can be calculated from the dielectric spectrum in Fig. 2 (c). The calculated resonance wavelength (λnes = 375 μm) is in agreement with the resonance frequency of νH. Meanwhile, the magnetic field energy distribution indicates that the MM of number I work as a magnetic dipole along the X-axis, which triggers the dark mode νH at 0.8 THz along the Y-axis. To the MM from number II to number VII, the symmetry breaking divides the cut-wire into two parts: a longer portion right to the displaced symmetric axis and a shorter portion left to the displaced symmetric axis. Such a symmetry breaking results in a current flowing mono-directionally from the left lateral-side of TSRR to the longer-portion of cut-wire, shown in Fig. 4 (a). Thus, the left lateral-side and longer portion of cut-wire form an asymmetric VA with obtuse opening angle, which contributes to the νL mode. To the MM from number II to number III, another current flows mono-directionally from the longer-portion of cut-wire to the right lateral-side of TSRR. Such a current loop make the conductively coupled MM to be another asymmetric VA with acute opening angle, as dominates the νH mode. Since the asymmetric deviation leads to a longer metal loop of VA, the resonance frequencies of νL mode and νH mode occurs redshift in THz spectrum. In agreement with the magnetic field distributions shown in Fig. 4(b), the magnetic dipole momentum of νL mode is in the mirror symmetry of νH mode. This phenomenon is in agreement with the PIT effect of asymmetrically coupled MM [17]. As such, the transparency window is recognized as an interference of VA, which leads to the observed slow light effect. To the MM from number IV to number VII, a counter-clockwise circulating current flowing along the metal loop of TSRR gradually take the place of above mono-directional current flows. Therefore, the origin of νH mode switches from high-frequency VA resonance to the LC resonance. Thus, the condition of destructive inference is damaged so that the transparency window becomes a crossover tails of the LC resonance and a low-frequency VA mode, which cannot creates slow light. A proposed explanation for this result is based on the switching of excitation pathway. To a cut-wire, the electric energy mainly concentrates on the two terminals not at the mid-point. Since TSRR is displaced along the cut-wire starting from the symmetric axis, the resonance excitation pathway switches from the electric field to the magnetic field, leading to the observed circulating current flows in TSRR. Such a change of excitation pathway transfers the PIT-effect to be PIT-like effect.Fig. 4 (a): Surface currents of the symmetry broken MM at the mode of νL and of νH. (b): Magnetic field distributions of symmetry broken MM at the mode of νL and of νH. Color bars: The relative strength of currents and magnetic energy.
In addition to give detailed insight into the localized slow light phenomenon of our symmetry broken MM, the spatial distributions of electric energy density of νL and νH modes are illustrated in Fig. 5, correspondingly. To the νL modes, the electric energy accumulates at the left-arm of TSRR and the right-end of the cut-wire. As such, the symmetric broken MM plays the role as asymmetric VA with two different lateral-side lengths. The horizontal polarized THz pulse excites the anti-symmetric mode of VA, of which the charges of opposite signs accumulate at both extremities of the VA contributing to a stronger restoring force, and a frequency redshift occurs. To the νH modes, the electric energy distribution is more complex. To the MM of number I, the energy concentrates on the two arms of TSRR and the two terminals of the cut-wire symmetrically. In this case, the MM works like a VA excited at symmetric mode. When a slightly asymmetric deviation (δ = 5 μm and δ = 10 μm) introduced into the MM, the electric energy on the left-arm of TSRR vanished, in accompany with the elimination of energy on the left terminal of cut-wire. Such a destruction of energy balance results in another VA of acute open angles consisted in the right-lateral side of TSRR and longer portion of cut-wire, as shown in the insets of Fig. 5. A further increase of δ results in redistribution of electric energy since most charges accumulate at the gap area as well as the two arms of TSRR. In agreement with the counter-clockwise circulating current shown in Fig. 4 (a), the νH mode switches from VA resonance to the LC resonance, and the gap plays the role as a capacitor. The asymmetric deviation increasing does not change the gap space. At this point, the resonance frequency of LC mode cannot be changed dramatically so that the frequency redshift of νH mode stops. Besides, a longer portion of cut-wire results in the energy loss so as to weaken the strength of LC oscillation. To the PIT phenomenon, it is approved that the side-modes appears to be the same resonance mechanism. Thus, a slow light effect can be observed. To a PIT-like effect, the side-modes originate from different type of resonances. Naturally, the transparency window is a crossover of the spectral tails of different resonance modes, which cannot support a giant dispersion to generate a slow light effect. In agreement with the simulation shown in Fig. 3 (c), a slightly symmetry broken (2 μm < δ <13 μm) results in a destructive inference of VA-mode so that a slow light effect is achieved. However, a further increase of asymmetric deviation (δ>13 μm) switches the excitation pathway so that the slow light phenomenon disappears.
Fig. 5 Electric field distributions of symmetry broken MM at the mode of νL and of νH. Color bars: the relative strength of electric energy. Insets: The gold symbols refer to the equivalent V-shaped plasmonic antenna (VA).
4. Conclusion
In summary, a localized terahertz slow light effect is observed in a symmetry broken MM made of conductively coupled dark resonators of identical frequency. One resonator is a TSRR and the other is a cut-wire dipole oscillator. A horizontal displacement between the TSRR and cut-wire introduces an asymmetric deviation δ into the MM. Then, a transparency window in between dual modes resonance is observed. To a slight asymmetric deviation (2 μm<δ<13 μm), the lateral-side of TSRR and portion of the cut-wire forms a couple of asymmetric VA, which results in a PIT. A maximum 25.9 ps group-delay is observed at the central frequency of the transparency window. However, the simulation result predict the maximum group delay can reaches 27 ps at δ = 8 μm. For a larger asymmetric deviation (δ>13 μm), the high frequency side-mode switch from VA-resonance to LC-resonance so that the condition of PIT-effect is damaged. Alternatively, a PIT-like behavior dominates the dual side-modes. The slow light effect vanishes in this region. As such, the slow light is localized only in the area of the destructive inference of VA modes. Aforementioned discovery will be valuable in creating a speed modulator of slow light for the application of THz telecommunication.
Funding
National Natural Science Foundation of China (Grant No. 61307130); Joint Research Fund in Astronomy (Grant No. U1631112) under a cooperative agreement between the National Natural Science Foundation of China (NSFC); Chinese Academy of Sciences (CAS) (Grant No. XDB04030000); Innovation Program of Shanghai Municipal Education Commission (Grant No. 14YZ077); Project of Science and Technology Commission of Shanghai Municipality (Grant No. 16695840600).
Acknowledgments
Z.Z. acknowledges the Innovation Program of Shanghai Municipal Education Commission (Grant No. 14YZ077). W.P. acknowledges the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB04030000). Z. L. acknowledges the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 16695840600).
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