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Abstract
Metastructure analogs of electromagnetically induced transparency (EIT) provide a new approach for engineering realizations of nonlinear optical manipulations regardless of harsh conditions; further can be employed in polarization conversions for its low-loss transmission and phase modulation. In this work, dual-band EIT in a dielectric-metal hybrid metasurface achieved via providing different coupling channels is theoretically investigated with a maximum group delay of 404 ps. The linear-to-circular polarization conversion (LCPC) behaviors are observed respectively holding the transmittance of 0.58 at 0.68 THz, 0.73 at 0.76 THz, 0.61 at 0.90 THz, 0.53 at 0.99 THz, owning to the asymmetric EIT responses in the transverse magnetic (TM) and transverse electric (TE) modes incidence. On the other hand, phase-transition VO2 is doped to perturb the dark mode resonances. With its conductivity σ = 105 S/m, dual transparency peaks transform into unimodal broadband transmission windows with relative bandwidths of 17.1% and 9.1% under the TE and TM excitations apart. Induced LCPC possesses a bandwidth of 10.4% centered at 0.76 THz attributed to the drastic dispersion. The as-proposed design exploits pattern asymmetry of EIT responses to realize LCPC, promising the wide prospect of reconfigurable multiplexings.
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1. Introduction
Electromagnetically induced transparency (EIT) ranks a coherent process rendering opaque media transparent with conspicuous dispersions, originally observed within an absorption envelope in a three-level quantum system [1]. As both radiative transitions generate a dark-dressed superposition without electric dipole induced [2], which give rise to the destructive interference accompanied by nonlinearity, the EIT effect has been universally employed in photon deceleration and storage [3,4], pulse regulation [5], and other nonlinear systems. The advent of metamaterial makes the harsh experimental handling [6] of EIT negligible, due to the subwavelength dimension-dependent meta-atoms interactions, demonstrated first in the dual-twisted trapped mode by Papasimakis et al. [7], functioning as a superior mimic pathway. In recent times, typical EIT metastructures (MST) are implemented through the interaction between both radiative and dark meta-atoms [8,9], as well as the structural asymmetry in multisystem [10,11]. Zhang et al. [12] novelly proposed a pseudo-complementary isolated metal meta-atom architecture, enhancing the localization of the magnetic field to achieve opaque background regardless of incidence sensitivity, which is further noted by Li et al. [13], expanding the generating pathway. In contrast with the lossy plasmonic resonance raising high dissipation in these metal mesoscopic cells, Mie resonance in dielectric MST is promising in accordance with the surpassing Q-factor and lossless transmission [14] confirmed by all-dielectric mode interferences [14,15] or the hybrid MST mimic [16]. As a solution to the transparent bandwidth confinement in particular frequencies, Kurter et al. [17] verified the feasibility of multimodal propagation by the overlapping bull's eye MST, which inherently offers the different coupling pathways based on eigenfrequencies despite the fixed status, can be further explored in tuned-materials doped cases [13,18].
On another side, the polarization manipulations of electromagnetic waves (EWS) based on MST, likewise, have attracted attention in this decade for the ultrathin thickness and geometry-dependence, promising for compact antennas [19], derived radar-phased arrays, and detectors for the military industry. However, high loss attributed to poor transparency in conversion procedures [20] acts as a bottleneck due to the opaque propagation of EWS. Besides, the reported typical polarization converters possess a stable phase difference, shrinking the further application in phase-mutation triggered delay devices. It is worth noting that Zhu et al. [21] innovatively established an EIT-based linear-to-circular polarization conversion (LCPC) MST, due to the asymmetric responses of both transverse magnetic (TM) and electric (TE) incidences, regarded as a surpassing frequency-selective MST associated with high group delay and low transmission loss. This cascade MST indicates a new path to the polarization controlling with slow-light characters, despite destitute frequency coverage and reconfigurability.
In this case, an LCPC comb tailored via dual-band asymmetry-mode metal-dielectric hybrid EIT is elaborated intrinsically to polish the multiplex ability in the information exchanging, with the maximum group delay over 400 ps, and the transmission peak over 0.85. Converted circularly polarized output with the corresponding transmission respectively located at 0.68 THz with 0.58, 0.76 THz with 0.73, 0.90 THz with 0.61, and 0.99 THz with 0.53, supported by 3-dB AR bands. The phase-transition VO2 is doped to enhance the reconfigurability. The unimodal broadband EITs in both TE and TM modes (taking 0.8-transmission as the standard, TE and TM EITs separately hold the relative bandwidths of 17.1% and 9.1%) are observed with VO2 slices in the metallic phase (conductivity σVO2 = 105 S/m), verified by the coupled oscillators system, while its insulating state (σVO2 = 10 S/m) contributes negligibly. The as-proposed multitasking device can be equipped in radomes, retarders, and other terahertz (THz) multiplex communication scenarios with significant potential. Compared to the EIT-based LCPC that has been realized in Ref. [32], the EITs in this work are established by the interactions between the dielectric Mie mode, the metal quasi-dark/radiative modes, and the transparency windows up to two (demanding high requirements for suppressions of mutual interference between coupling paths). Besides, the asymmetry of EITs mainly depends on the different electric sizes and the different dressed modes in both polarizations with superior linear shapes rather than the topological rotation (as shown in Ref. [32]). Besides, with the doped VO2 (which can be grown with methods of direct current magnetron sputtering and pulsed laser deposition [34]) changing from an insulator to the metal, the dual transparency windows transmit to the wideband ones under both modes, and the four-frequencies LCPCs gradually turn into the broadband LCPC with remarkable modulation intensity. Finally, it should be emphasized that our design is predominantly a theoretical research, the fabrication technology is not the focus.
2. Theoretical model
The ultimate periodic principle and the unit cell of the configuration are elaborated in Fig. 1. The unit (see Fig. 1(b)) can be divided into three layers for further narration as shown in Fig. 1(c), where double titanium dioxide (TiO2, εTiO2 = 114, loss tangent tanδTiO2 = 0.00027) [22] rings topologically placed against the top dielectric-1 (εdielectric-1 = 3.794, tanδdielectric-1 = 0.003, equivalent to the fused quartz operating at 60 GHz, it can be synthesized by the microdoped fused quartz, processed by laser-ablation) [23,35], consisting the first layer. A y-oriented gold (conductivity σgold = 4.56×107 S/m) [24] cross-cutting wire (GCW) is sandwiched in the quartz background (deposited by E-beam evaporation) [34], regarded as the second element. Dual-split-rings (DSS) with another GCW etched in parallel on the bottom dielectric plate.
Fig. 1. Schematics for the proposed reconfigurable EIT-based LCPC MST with the periodic arrangement and unit-cell. (a) The periodic topological arrangement of the MST with the unit-cell extracted at the bottom right, LPx and LPY respectively represent the x-linear (TM) and y-linear (TE) polarized lights, CP refers to the converted circular polarized light. The stereoscopic view (b) and triple-layers sectional view (c) for the model.
In this analog, the wave vector is set along the z-axis. Both electric and magnetic fields distribute parallel to the x- and y-directions with periodic boundaries. Consequently, TE and TM modes are equivalent to the x-and y-linear polarizations, the final detailed geometric parameters are integrated into Table. 1. Here, the evolutionary procedure can be divided into two iconic stages (1 and 2). In Stage 1, the superior dual-peaks EIT effect is configured via the destructive interference between the GCW along the y-axis (radiative electric dipoles mode), dual-TiO2 dielectric rings (DDR, dark Mie mode), and the electric DSS (regarded as the quasi-dark mode) under the TE incidence is illustrated firstly, with four-level tripod system established to analyze the inside physical mechanism, where the original geometric dimensions are the same as shown in Table. 1 except the initial radius of the DDR r = 21 µm, and the length of the y-oriented GCW l1 = 110 µm. When in Stage 2, concentrations are poured on generating the TM excited dualband EITs at the beginning. Given the radiation characteristics (of the GCW), an x-oriented GCW with the original length l2 = 110 µm, the width wx of 10 µm is further introduced coplanar with the DSS, with other parameters unchanged. Then, in order to create a suitable phase difference (to generate the LCPC), we set l1 = 115 µm l2 = 105 µm, and the width wx = w1 = 8 µm (unified with w1 finally) to enhance the asymmetry of respective EITs (in both polarizations), defining r = 23 µm comprehensively taking account of the transmittance and the converted bandwidth (other par). VO2 films are ultimately doped to improve the reconfigurability of this design. Next, this process and phased results will be elaborated theoretically.
Table 1. Comprehensive geometric parameters for the final equipment
3. Analysis and discussion
3.1 Stage 1: the dual-band EIT MST excited via the TE mode
Figure 2 provides the initial deployment (components can be extracted from Fig. 2(a)) comprising of the DDR, y-oriented GCW, and DSS, isolated by the equal-thick quartz bases (thickness h = 5 µm) accompanied with corresponding transmission responses. Irradiated by the TE waves (see Fig. 2(a)), the dual-peak EIT is observed with both transmission peaks PI = 0.93, PII = 0.96, separately locate at 0.73 THz and 0.94 THz, as well as the transmission dips frequencies fdI = 0.64 THz, fdII = 0.88 THz, and fdIII = 1.05 THz. For a deep insight into the physical formation mechanisms in this system, transmission responses of each resonator (the DSS, DDR, GCW) are listed separately in Fig. 3, with respective distributions of the induced electric field at resonance frequencies.
Fig. 2. (a) The exterior of the device illustrated from the top, bottom, and the stereoscopic view apart. (b) The transmission responses (amplitude with two peak PI, PII, and the dips dI, dII, dIII, with the changing phase).
Fig. 3. TE-mode-excited transmission amplitudes of each element with related electric field distributions at resonant frequencies. The transmission response for (a) the middle y-oriented GCW, showing the resonance at 0.71 THz (fa), (b) the DDR, indicating the resonance at 0.96 THz (fb), and (c) the DSS, possessing the resonance at 0.99 THz (fc). (d), (e) Transmission spectra for the combined dual-peak EIT MST, where peaks Pa in (d) and Pb in (e) are two transparency frequencies highlighted by the red dotted lines, symbols Da, Db, and Dc respectively represent transmission dips (consistent with that noted in Fig. 2(b)). (f)–(h) Exhibit the individual electric field distributions at resonance frequencies (fa, fb, fc), with the sectional views of the integrated MST displayed in (i) at the 1st transparent frequency fpa = 0.73 THz, and (j) at the 2nd transparent frequency fpb = 0.94 THz. (Longitudinal sections of the electric field are shown with the corresponding model diagram.)
With TE polarization excitation, the GCW (along the y-axis) resonator holds a wide Lorenz linear envelope at fa (see Fig. 3(a)), in contrast with the shallow sharp transmission dips at fa in the DDR resonator (shown in Fig. 3(b)). Figure 3(c) notes a slightly narrow dropping profile in the transmission of the DSS, compared with that of the GCW. Here employing quality factor Q = f/Δf as the fundamental of the basic resonant mode (f and Δf respectively refer to the resonance frequency and half-maximum bandwidth respectively), the isolated GCW (QGCW = 2.54) resonator can be demonstrated as the radiative mode, and the DSS (QDSS = 12.91 ≈ 5QGCW on the same scale) refers to a sub-radiation state, further regarding DDR resonator as the dark mode for its QDDR >> QGCW/QDSS in the TE mode. As shown in Fig. 3(f), the energy of the induced electric field is mightily confined on both ends of the GCW, operating as the electric dipole. From another side, the localization characteristics (of electric fields) around TiO2 DDR (see Fig. 3(g)) imply a Mie resonance at 0.96 THz, contributing to a pair of reverse magnetic dipole moments. Besides, the strong confinement of the electric field at the splitting slot in the excited DSS resonator intuitively demonstrates the resonance at fc = 0.99 THz, which can be considered as another electric excited radiative mode. Figures 3(i) and (j) comprehensively denote the electric field excitation at transparency frequencies (fpa = 0.73 THz, fpb = 0.94 THz). The confined field energy around the GCW and GSS overwhelms that in the DDR, which reveals a destructive interference in the bright-bright mode between the GCW and GSS. In detail, the energy located on the DSS ranks at a slightly higher level compared with that on the GCW, indicating its sub-radiation properties supported by the Q value (QDSS > QGCW). In other words, the coupling between the DDR and GCW resonators contributes mostly to the transparent peak Pb at fPb due to conspicuous energy localization. Notice the bottom DSS, where the induced electric field is partly distributed around two splitting slots (see the section view related to the DSS model in Fig. 3(j)), it can be equivalent to dual-metal short-wires, playing the role of the weak electric dipole with poor radiation characters (compared with the solitary excitation state) in this case. Therefore, Pb is dominantly triggered in accordance with a bright-dark-mode coupling.
Note total Q-factors relationship QGCW < QDSS ≪ QDDR, the DSS resonator can also be defined as the quasi-dark mode since it not only can couple with the external electric field directly but with the bright mode in general [25,26]. Under the TE-mode excitation, the DSS unit is straightly excited by the induced electric field, possessing the properties of the sub-radiative mode (as Fig. 3(c) confirmed), which differs from the TM mode response (that will be demonstrated later). To look into the formation principles in this EIT MST, the four-level tripod system is constructed as shown in Fig. 4.
Fig. 4. The four-level tripod system employed to denote the dual-band EIT response. Ωi and γi (i = 1, 2, 3) individually are the transition phase and the interstage damping factor, |a > can be classified as the ground state, |d > and |b > refer to the excited state and metastable state respectively, the sub-radiation level is symbolized by |c > . κ0 and κ partly correspond to the coupling coefficient between levels |a > and (|b>) |c > .
As illustrated in Fig. 4, the ground state |a > interacts with the excited state |d > under the excitation (the TE mode coupling) field irradiation, subsequently establishing one transition channel, where the radiative dipole-adapted transits with a Lorenz oscillation frequency. The coupling between the metastable level |b > and |d> (the excited state) gives rise to another pathway with phase detunings mediated via extra probing beams with coupling coefficient κ1 reacting level |a > . Supposing the quasi-dark properties exist, the coupling path between levels |b > and |d > can be supplementally represented as the dotted line, directly excited by the coupling field (i.e., the TE mode). Likewise, the third transition pathway |b> → |d > can only be established by coherent interaction between levels |a > and |c> (the sub-radiation level), owning to the stable incentive responses, mediated via an extra probing beam to compensate for phase detuning. Actually, the implicit coherent interference lies between channels |b> → |d > and |c> → |d>, whereas, weaken by the stronger coupling between pathways |a> → |d > and |c> → |d > on account of the high radiation carried by ground-state |a > at transition. with a reacting coupling coefficient κ2 despite the direct transition from levels |c > to |d > is forbidden.
Here, take account of quasi-dark characteristics for the DSS element. Assume the system is in the TE mode polarization (noticing the model coordinate system insets), the system will introduce three coupling ways (the direct: |a> → |d > and |b> → |d > in the TE mode, and the sideways: |a> → |d> → |c> → |d>). If the interference is destructive, the absorption in the wide spectrum will be suppressed heavily, opening up narrow transmission transparent windows, i.e., specific regions between levels |a > and |b> (|c>) will be loss-free, generating a dual-peak EIT behavior. This integrated MST effectively employs the radiative mode GCW to alternate the pathway |a> → |d>, the quasi-dark mode DSS to the channel lies between |b> → |d>, and the channel between levels |d > and |c > are replaced by the dark mode DDR component.
3.2 Stage 2: the LCPC based on dual modes irradiated asymmetrical dual-peak EITs
Aimed at the realization of the TM mode excited EIT to satisfy the synthesis conditions for circularly polarized waves symbolized by Eqs. (1) and (2), where tii represents co-polarization transmission coefficient, φii is the transmission phase [21], another GCW along the x-axis is placed at the bottom of the model taking the or (coplanar with the DSS), with geometric parameters: y-oriented width w1 = 8 µm, x-oriented length l2 = 105 µm. The improved deployment is displayed in Fig. 5, where the stereoscopic diagram from the side view is noted in Fig. 5(a) (the added GCW is outlined with the white dashed circle), and the bottom and top views are shown in Fig. 5(b).
Fig. 5. (a) Stereo-side view of the polished device, with the white-dotted circle emphasized the extra x-oriented GCW. (b) Bottom and top perspectives of the model.
The added GCW resonator operates as another radiative mode with TM-mode excitation, coupling with the coplanar DSS and the top DDR both act as the dark mode, supported by the comparison between the original TM-transmission response and that in the improved status shown in Fig. 6. The initial transmission response is described in Fig. 6(a), where a narrow shallow dip falls at 1.03 THz attributed to the dark Mie resonance in the TiO2-DDR, and the DSS here plays a dark mode role, (i.e., corroborating the quasi-dark mode properties of the DSS). The lack of the radiative mode blocked its direct transition to the excited state, which can be alternated by the increased GCW. Noted in Fig. 6(b), the coupling with the DDR and DSS synchronously generates the TM-excited dual-band transparency windows (transmission peaks PA = 0.90 and PB = 0.89), individually located at 0.79 THz and 1.0 THz. Symbols dA, dB, and dC correspond to three sharp transmission dips in the modified system. In this case, the TM mode excited dual-peak EIT effect is realized.
Fig. 6. The transmission-responses comparison. (a) Excited transmission amplitude (the red line) and phase (the blue line) versus the original integrated configuration under the TM-polarization. (b) Transmission responses with TM mode excitation (amplitude with the brown line, phase with the pink line) versus the improved device. Bottom-view insets of the corresponding state are illustrated at the bottom left.
As universally known, group delay (GD) is a keystone to measuring the EIT dispersion properties contributing to slow-light behaviors and confirmed via abrupt phases. According to Eq.(3) [27], where φ and f represent the transmission phase in single-polarization incidence and the operating frequency, respectively.
The derived results are displayed in Fig. 7(a), implying peak values P1 = 238 ps at 0.69 THz and P2 = 315 ps at 0.92 THz in the TE mode, Px = 342 ps at 0.77 THz, Py = 404 ps at 1.0 THz and Pz = 222 ps at 1.1 THz in the TM mode. Besides, the entire GD response can also be approximately regarded as the contributions of the interaction couplings raised discrepant GD components between the GCW and DDR, the GCW and DSS resonators. Figures 7(b) and (c) respectively provide the GD curves under both modes in the DSS-removed and DDR-removed systems. The GD peaks (P1 and PX) at lower frequencies mostly result from the coupling between the DSS with GCW (see Figs. 7(a) and (b)) while the peaks (P1 and PX) at higher frequencies are attributed to that between the DDR and GCW to a large extent (see Figs. 7(a) and (c)) in both modes, evidently consistent with the resonance properties of each element.
Fig. 7. (a) The deduced GDs in the integrated system according to the steep transmission phase, symbol Pi (i = X, Y, Z, 1, 2) represents individual maximums, in the (b) DDR-removed and (c) DSS-removed systems, under TE and TM modes. Corresponding models are individually illustrated with the insets from the top view (the left ones) and the bottom view (the right ones).
For a comprehensive analysis of the derivative polarization characters, dual modes (TE and TM) excited EIT responses are integrated, and the S-parameters are listed in Eqs. (4) and (5), where tii and Δφ are the co-polarization transmission coefficient and phase difference apart (as noted before in Eqs. (3) and (4)), the 3-dB axial ratio can be further deduced according to Eqs. (6) and (4) [21,28]. The entire outcomes are denoted in Fig. 8.
Fig. 8. (a) Transmission amplitudes with both polarization modes. (b) The phase difference with tolerable error ranges highlighted in yellow shadows, and (c) the computed 3-dB AR band.
Four common transmission frequencies fA = 0.68 THz, fB = 0.76 THz, fC = 0.90 THz, and fD = 0.99 THz, respectively possess the amplitudes of 0.58, 0.73, 0.61, and 0.53 as depicted in Fig. 8(a). The dual-modes phase difference Δφ is synchronously described in Fig. 8(b), satisfying the phase condition (referring to Eq.(2)) at listed common frequencies, where the typical LCPC effects are observed, verified via the AR values shown in Fig. 8(c).
The electric field distributions in this improved model are investigated for further understanding of the physical mechanism. Taking profiles at fA for example (given the essentially same situations at other frequencies), the induced electric fields extracted from different z-sections are revealed in Fig. 9 with TE- and TM-modes incidence.
Fig. 9. Electric field distributions on individual components viewed from corresponding z-sections in both TE and TM modes at fA (showing with 2×2 unit-cells). (a)-(c) Respectively correspond to the field profiles on (a) the DDR with section z = 5 µm, (b) the GCW with section z = 0 µm, and (c) the DSS with section z = -5 µm in the TE mode, while the same goes from (d) to (f) with the TM-excitation.
In the TE mode incidence at fA, the y-oriented GCW is directly excited operating as the electric dipole, with the induced energy is localized at both ends (shown in Fig. 9(b)), coupling with the top DDR (see Fig. 9(a)) and the bottom DSS (noted in Fig. 9(c)) resonators via different pathways, generating two transparency windows. It is worth noting that the DSS also can be excited in this status The electric field energy is confined mostly around two arms in the horizontal direction and the gap of the single split-ring (shown by Fig. 9(c)), simultaneously interacting with the middle GCW (along the y-axis) and the top GCW, confirmed by the distributions in the mapping locations (sections z = 0 µm and z = 5 µm) of the bottom DSS. The energy intensity in section z = 0 µm equivalent to the superimposition between the radiative GCW and the coupling from the bottom DSS, ranking at the highest level (compared with other sections). Likewise, the horizontal GCW (in Fig. 9(c)) is activated in the TM mode, coupling with the dark-state dressed bottom DSS and the top DDR. Besides, the y-oriented GCW is non-radiative which differs from the TE-excitation case (Figs. 9(a)-(c)). In general, radiative modes can be excited from both polarizations at the common frequency fA, forming asymmetrical EITs, which rigorously verified the EIT eliciting mechanism in the LCPC.
To enhance the reconfigurability of this equipment, the phase-changing VO2 films at doped at both gaps of single bottom split-rings in the DSS, given the resonance-frequency driven transparency intensity and locations in the spectrum. The dielectric permittivity of VO2 can be described through the Drude model as Eq. (8) [24]:
Fig. 10. (a) The bottom and top views of the VO2-doped device, (the red part is the VO2 fillings at the bottom with the dotted circle for emphasis). (b) Transmission amplitude with VO2 in the metal state (σVO2 = 105 S/m), presents the classic EIT linear shapes in both TM and TE incidence, symbols PE/PM and dEi/dMi (i = 1, 2) individually refer to the TE/TM-excited transparency peak and transmission dips, f0 = 0.76 THz is the common transmission frequency with the amplitude of 0.55.
With metal-phase VO2 padding in the gaps (shown by Fig. 10(a)), typical unimodal EIT behaviors are noticed under both incidence modes as elaborated in Fig. 10(b), the TE transparency peak PE = 0.97 at fPE = 0.88 THz, the TM transparency peak PM = 0.92 at fPM = 0.98 THz, with corresponding transmission dips respectively symbolized by dE1, dE2 and dM1, dM2. Coincidentally, the transmission peaks in one polarization mode correspond to the transmission valleys (dips) in its orthogonal mode, for PE = 0.97 versus dM1 = 0.06, fPE ≈ fM1 (f represents frequency), PM = 0.97 versus dE2 = 0.10, fPE ≈ fM1. Therefore, the polarization separation effect can be realized with VO2 working in the metallic state. In addition, the TE-mode transmission coefficients over 0.8 span from 0.80 THz-0.95 THz with a relative bandwidth (RBW) of 17.1%, while the satisfied region in the TM mode extends from 0.94 THz to 1.03 THz, RBW = 9.1%. Further analysis reveals the EIT-based LCPC phenomenon at f0 = 0.76 THz, verified with the dual-modes phase difference and the deduced AR band as provided in Fig. 11.
Fig. 11. With σVO2 = 105 S/m, (a) the phase difference, where the yellow shadows are employed to highlight the tolerance-error regions, and (b) the computed AR response, f0 = 0.76 THz.
The phase difference with dual modes excited EIT responses is exhibited via Fig. 11(a), where the condition (Eq.(2)) is met at f0. The derived 3-dB AR band (see Fig. 11(b)) further indicates that the operation spectrum centers f0 and extends to both sides with a decaying magnitude attributed to the strong dispersion, spanning from 0.73 THz to 0.81 THz, RBW = 10.4%. Besides, a narrow conversion band emerges at 1.02 THz (shown in Fig. 11(a)), for the resembling conversion principles, accordingly, the LCPC effect is achieved at dual bands different from the non-doped device.
GDs are rededuced in Fig. 12, where dual-modes excited GDs individually possess three maximums, the TE-excited peaks PI = 111 ps at 0.79 THz, PJ = 203 ps at 0.98 THz, and PK = 252 ps at 1.01 THz, while in the TE-illumination PR = 187 ps at 0.93 THz, PS = 192 ps at 1.06 THz, and PT = 303 ps at 1.10 THz ranks the largest, which implies its polarization-sensitive slow-light properties. Admittedly, the doped VO2 changes the resonance frequency of the bottom GCW according to the equivalent circuit theory, with identical dark-modes resonance frequency, the coupling path will decrease, and the dual-peak EIT degenerates into the classic single-band EIT accordingly. Whereby, with the regulation of σVO2, the transition from the dual-modes EITs to the broadband unimodal ones, as well as the based multiband LCPC to the broadband one.
Fig. 12. With σVO2 = 105 S/m, the computed GDs from both TE and TM modes, local maximums respectively symbolized by PI, PJ, PK (in the TE mode), and PR, PS, PT (in the TM mode).
3.3 Parameters discussions and fittings
Given the crucial factors affecting ultimate EITs, both decisive GCW impacts the entire linear shape, i.e., the coupling intensity, due to the high radiation loss, excited by corresponding polarization modes (the induced electric field parallel to the elongation direction). Here, DDR and DSS resonators are both involved in the destructive interference under TE/TM incidence, related geometric dimensions supposed to be poured attention to. Besides, polarization stability also remains a concern, and the transition procedure demands to be further refined.
Here, adjusting σVO2 (i.e., parameter σ) from a value of 10 S/m (the insulation state) to 105 S/m (the insulation state), transition progress is first provided in Fig. 13. With σ growing gradually, (displayed by colored dotted lines marked TE transmission in Fig. 13(a)) the middle transmission dip rises and the remaining two gather towards it, finally generating a broadband EIT (RBW = 17.1%) in the TE mode. For the case of TM incidence (see the responses with colored solid lines), the original transparency window at low frequencies decays, replaced by a brand new transmission valley, i.e., dips at low and middle frequencies of the response spectrum combining into a single zero-transmission interval, while transparency region at high frequencies extends to a final RBW of 9.1% in line with increased σ. For another, when σ = 103 S/m (shown by the blue dotted line in Fig. 13(b)), one of the initial LCPC at f2 transforms into the normal linear polarization transmission, which reverts to circularly polarization output again with σ continuing to rise (σ over 104 S/m), and the LCPC effect is observed at a brand new frequency f5. By contrast, no LCPC effect occurs at the other three original frequencies f1, f3, and f4 with σ over 5×103 S/m (see the brown-line highlighted AR values over 1.5 dB in Fig. 13(b)). The f2 centered narrow conversion band holds an RBW of 10.4% when VO2 in the metallic phase ultimately (σ = 105 S/m), attributed to the dispersion in EIT effects. Supplementally note that the separate LCPC at 1.08 THz (around f5) corresponds to an amplitude of 0.37 (σ = 5×103 S/m), and the new LCPC emerges at 1.02 THz (see the high-frequency part of the AR band indicated by the purple solid line with σ = 105 S/m) with an amplitude of 0.72. Overall, the reconfigurability of the system is significantly enhanced. Then the stability of the device is rigorously demonstrated via modifying the polarization angle α spanning 0°-45°, as exhibited in Fig. 14.
Fig. 13. (a) Transmission responses versus σ, TE transmission is symbolized by colored dotted lines, TM-transmission by colored solid lines. (b) The corresponding AR values, with σ = 10 S/m, f1 = 0.68 THz, f2 = 0.76 THz, f3 = 0.90 THz, f4 = 0.99 THz are four LCPC frequencies in the insulation state, f5 = 1.08 THz refers to a brand new LCPC frequency when σ = 103 S/m (the fluctuation range of σ:10 S/m ∼ 105 S/m).
Fig. 14. The transmission responses versus α ranging from 0°-45° on the premise of σ = 10 S/m, (a) α here refers to the angle between the electric field vector and the y-axis, investigating the stability of the TE-EIT response, (b) α, in this case, is the induced angle between the electric field vector and the x-axis, investigating the stability of the TM-EIT response.
It can be found that the EIT behaviors in the MMS present poor stability due to the significantly reduced transmittance and the heavily destroyed line shape under both incidence polarizations with α over 15°. In the same case, triggered LCPC effects also revert to normal linear-polarization transmission constrained by harsh phase conditions (the AR bands will not go into detail). Comparison (between TE and TM transmission responses) further reveals the congruent transmission amplitude in both TE and TM modes when α = 45°, on account of the field component along the x-axis equaling to that along the y-axis. Here the transmission responses with VO2 in the metal state are omitted for its possessing the identical varying trend (that working in the insulating state).
Further, the radius r of the top TiO2 rings and the length l2 of the middle x-oriented GCW are investigated to precisely master the impacts of the geometric dimensions on the EIT line shape. Transmission responses with related AR values versus r and l2 are noted separately in Figs. 15 and 16.
Fig. 15. The transmission responses versus r with computed AR values, (a)-(c) are in the insulating mode with σ = 10 S/m, (b)-(c) are in the metal state with σ = 105 S/m, (a) and (d) correspond to the TE mode excited transmission amplitude, while (b) and (e) correspond to the TM mode excited transmission, (c) and (f) respectively refer to the AR bands in the TE mode and TM mode.
Fig. 16. The transmission responses versus l2 with computed AR values, (a)-(c) are in the insulating status with σ = 10 S/m, (b)-(c) are in the metallic state with σ = 105 S/m, (a) and (d) correspond to the TE mode excited transmission amplitude, while (b) and (e) correspond to the TM mode excited transmission, (c) and (f) respectively refer to the AR bands in the TE and TM modes.
The transparency window in the high-frequency region renders a redshift accompanied by transmittance decays, syncing with the growing r under both TE and TM modes with σ = 10 S/m (see Figs. 15(a) and (b)). Herein, the two high-frequency LCPC locations synchronously exhibit the red-shifted behavior verified by the AR band in Fig. 15(c), in contrast to the stable state of the conversion frequencies at low frequencies.
As illustrated in Figs. 15(d) and (e), the transition trend is consistent with that operating with σ = 10 S/m. Besides, the TE transmission dip at high frequency (see Fig. 15(d)) possesses a wide range of variation compared to that in the TM mode (shown by Fig. 15(e)). Likewise, the redshift effect is observed in computed AR bands in Fig. 15(f). r = 23 µm is the set final value comprehensively taking account of the transmittance and the conversion intensity.
The disturbance of the variable l2 on the TE mode transmission can be ignored with VO2 operating in both states as shown in Figs. 16(a) and (d), owning to the extension direction of the added GCW. On the contrary, the x-oriented GCW is directly excited under the illumination of the TM-polarization, with its dynamic electric dimension introducing the perturbations in corresponding EIT responses verified by Fig. 16(b) and (e). Frankly, the obvious redshift of the zero-transmission frequencies in TM dual-band EIT is captured syncing with the growing l2, (see Fig. 16(b), σ = 10 S/m), while the transparency regions remain stable. Therefore the deduced AR bands merely fluctuate in the conversion intensity (the AR values) rather than in the frequencies-intervals. When VO2 operates in the metallic status (see Fig. 16(e)), the transparency bandwidth in the TM mode expands as l2 increases, further triggering variations in LCPC bandwidth and the transmittance of the circular polarization light shown by Fig. 16(f). l2 is set as 105 µm after comprehensive consideration. Other geometric parameters are listed in Table. 1.
As is widely acknowledged, this type of dual-band EIT, i. e., the EIT responses with VO2 acted as an insulator (the transmission responses in both modes are integrated into Fig. 8(a)) can be numerically analyzed with the triple particle model. The system can be represented with the following equations [31]:
Assuming labels 1, 2, and 3 respectively symbolize the GCW (the bright mode), DDR (the dark mode), and DSS (the quasi-dark mode) resonators. In this case (x1, x2, x3), (ω1, ω2, ω3), and (γ1, γ2, γ3), are respective displacements (possessing the complex form xi = Niejωt, a function of angular frequency ω), damping factors, and resonance frequencies of oscillators. The coupling strengths between the GCW and the DDR, the GCW and the DSS are characterized by symbols Ω1 and Ω2 apart. (M1, M2, M3) and (Q1, Q2, Q3) represent respective effective charges and masses. E = E0ejωt is the incident electric field.
The magnetic susceptibility can be written with the following formations by solving the Eqs.(9)-(11) [31], Where P is the polarization of the incident waves. Parameters ai, Di, ξ, η (i = 1, 2, 3, 4) are all the multivariate functions of ω, ωi, Qi, and Mi (i = 1, 2, 3).
It is obvious that numerical results can be yielded via further fitting (transmission T = 1-imag(χ)). However, the progress is complex and cumbersome, meanwhile, the coupling between the DDR and the DSS excited by the TE mode is expected to be taken into account due to the quasi-dark properties of the DSS (in this case, it can be considered as a bright mode with a Q factor comparable to that of the radiative GCW resonator), compared with the design in Ref [31]. This particular state makes the fitting conditions more complicated. It should be noted that one particular physical model is to numerically verify the resonance modes and the relevance that we have gained in the simulation, (e.g., the damping factor of the bright mode is expected to be much larger than that of the dark mode). Actually, the triple particle model can be further simplified.
Assume the coupling between the DDR (vibrator A) and the GCW (vibrator B) generates at first, there will also be a typical EIT response. In this situation, define a new vibrator to symbolize the state synthesis of vibrator A and vibrator B. Then the new vibrator couples with the DSS resonator (vibrator C) further generating the EIT response. Hence, the triple particle model can be simplified to the dual particle model and the essence for the generation of the EIT is invariable, employing different models aiming at finding the coupling relevance. Here with VO2 operating in the metallic state, the classic broadband EITs are observed under both modes, it can be understood that resonance frequencies of the DSS and DDR are overlapping, corresponding to the overlap of coupled channels, and the dual particle model is overall suitable (utilizing the bright and dark vibrators to distinguish due to their different radiation levels). In the dual particle model (corresponding to σ = 10 S/m), the kinetic differential equations are presented as:
In this case, the deduced fitting parameters are: β = 0.329, Ω = 0.461, γ1 =0.213, γ2 = 0.021 in the TE mode incidence, β = 0.290, Ω = 0.177, γ1 = 0.175, γ2 = 0.025 in the TM mode incidence. Note γ1≫γ2 under in both polarizations, implying a drastic radiation dissipation in the exciting GCW resonators, the analytically derived transmission traces are depicted in Fig. 17, compared with the simulation results. Tolerable errors can be attributed to the transmission loss and the iteration numbers with taking approximations. Generally, the established dual-vibrators system reproduces the EITs validly, elaborating inside physical mechanisms equivalent to the role of a three-vibrator, verifying the feasibility of the system in a more convenient way.
Fig. 17. The comparisons between the EIT transmissions by simulation analysis (the red traces) and analytical calculations (the blue traces) under (a) the TE and (b) TM polarization incidence.
Finally, for a systematical and intuitive novelties-impacts description of this design, the related EIT devices with superior behaviors reported earlier are summarized, which are listed in Table 2. for comparison. Here in the column “Operation performances and frequencies”, GDm and Tm respectively refer to the maximum group delay and the maximum transmittance of the transparency windows, RBWCPM and TCPM are the maximum converted relative band and transmittance of the circular polarization respectively. Besides, a reconfigurable device can be adjusted via contactive (e.g. extra bias-voltage) or contactless (e.g. the pulse/magnetic/gravity fields, or thermal control) means, which is more flexible and convenient in operations (compared with the former). Obviously, the operation performance of this thermal LCPC switch based on reconfigurable EITs is advanced and valuable, which is better than those mentioned in the past overall.
Table 2. Comparisons between this work and reported EIT devices devices
4. Conclusion
In summary, the asymmetrical dual peak EIT responses are theoretically reproduced and numerically demonstrated under the TE and TM modes in the dielectric-metal hybrid MST, which provides muti-channels coupling generated destructive interference with maximum GD reaching 404 ps. The DDR and DSS elements respectively present radiative electric dipoles resonance and the Mie mode resonance, while the DSS resonator plays a role of quasi-dark mode for its different excitation properties in both polarizations. The LCPC effects are captured individually with the transmittance of 0.58 at 0.68 THz, 0.73 at 0.76 THz, 0.61 at 0.90 THz, and 0.53 at 0.99 THz verified by the 3dB AR bands. With reversible phase-transition VO2 flakes doped, the dual-band EIT behaviors transmit to the asymmetric broadband unimodal EITs with respective RBW of 17.1% (TE mode) and 9.1% (TM mode) confirmed via the dual-coupling oscillator system when VO2 operating in the metal phase (σ = 105 S/m), which further gives rise to the broadband LCPC centered at 0.76 THz, attributed to the superior dispersion, and the maximum GD approaches 303 ps. This novel reconfigurable EIT-associated polarization manipulation configuration has wide promise for antenna arrays, radomes, retarders, and multiplexing scenarios, supplying new pathways for novel multitasking thermal-magnetic switch designs.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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