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Abstract
We present a dynamic metamaterial based on Dirac semimetals and capable of realizing broadband and tunable asymmetric transmission in the terahertz region. The Dirac semimetal resonators have a chiral structure patterned with double-T resonators that results in partial polarization conversion of waves incident upon the material, leading to asymmetric transmission across a wide frequency range. We show how the gradual shift of the semimetal Fermi energy permits a method of control over the asymmetric total transmission.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Asymmetric transmission (AT) is the phenomenon in which an electromagnetic wave experiences different propagation behavior based on the direction of its travel through a medium. AT has become an increasingly researched topic due to its potential applications in communication and information transmission [1–3]. Conventional approaches for asymmetric transmission include using magnetic-optical materials [4], nonlinear materials [5] or time-dependent media [6], but these are not conducive to on-chip integration due to their larger size or impractical requirements (e.g. high power). AT may also be manifest when a wave of a particular polarization experiences a partial polarization conversion whose magnitude depends on the direction of propagation through the media. Recently, chiral metamaterials have attracted much attention due to their demonstrations of optical activity, such as circular or elliptical dichroism [7], negative index [8] and asymmetric transmission [9,10]. In 2010, Menzel et al. [11] reported asymmetric transmission for linearly polarized waves in a three-dimensional chiral optical metamaterial. Since then, chiral metamaterials have been widely used to achieve asymmetric transmission of linearly polarized waves in the microwave [12] and THz [13] regions. For example, Feng et al. [14] proposed a chiral structure composed of 90° twisted split ring resonators to achieve AT. Shi et al. [15] demonstrated a chiral ultrathin 90° twisted Babinet-inverted metasurface to realize a broadband AT with transmission contrast better than 17.7 dB. Mirzamohammadi et al. [16] presented a novel bi-layered chiral metamaterial to realize high–efficiency broadband AT that also exhibited strong polarization conversion efficiency. However, most of these devices were made of metallic resonators, tuned to different wavebands by carefully optimizing the geometric parameters [17–19], which significantly constrains certain applications.
Tunable devices continue to attract much research interest by enabling dynamic behavior, such as frequency tuning, stimulated by externally changeable conditions. Many tuning mechanisms exist, such as voltage application, temperature control, and chemical doping. These may be applied to various interesting materials including graphene [20,21], MoS2 [22], Si-based all-dielectric metamaterials [23], and VO2 [24]. Three-dimensional (3D) Dirac semimetals (DSMs), also known as three-dimensional analogues of graphene, have recently attracted much attention in physics and materials science because of their extremely high mobility, ultra-fast transient time and low-energy photon detection [25,26]. DSMs have several favorable properties, including greater stability in ambient environments [27], higher Fermi velocity [28] and carrier mobility [29], and easy tunability of Fermi level [30]. DSMs are relatively easy to manufacture and have lower intrinsic loss and more stable physical properties in terahertz band. Importantly, the dielectric constant function of DSMs is dynamically controllable by changing the Fermi energy, which can occur by alkaline surface doping or application of a gate bias [31,32]. Compared with graphene (2×105 cm2V−1s −1, 5 K) [33], DSMs have higher carrier mobility (9×106 cm2V−1s −1, 5 K) under the same conditions [34]. Research leveraging the tunable properties of Dirac semimetals has also begun to generate widespread interest. Wang’s group proposed a narrowband absorber based on a Dirac semimetal at terahertz frequencies [35], a double-layer bulk Dirac semimetal (BDS) sheet waveguide [36], and a deep-subwavelength BDS-insulator-metal waveguide [37] for manipulating terahertz surface plasmon polaritons. Zhang’s group proposed tunable polarization-insensitive electromagnetically induced transparency [38], plasmon-induced transparency [39] and linear-to-circular polarization conversion [40] based on DSMs at terahertz frequencies.
In this paper, we propose a planar chiral material which achieves AT of linearly polarized waves in the terahertz region. Unlike previous work, the structure is based on DSMs, meaning it can be actively tunable in its asymmetric behavior. The structure is composed of two double-T resonators on both sides of a dielectric layer. The asymmetric transmission is dynamically controllable by changing the Fermi energy of the DSMs. We present the structure and its AT performance with its polarization conversion behavior. This work offers another step in developing tunable AT for applications in detectors, communications and general polarization-sensitive electromagnetic devices.
2. Structure design and theoretical analysis
Figure 1 illustrates the unit cell of the designed chiral metamaterial, which consists of two DSMs-based double-T resonators printed on opposite sides of a polyimide substrate. The structural continuity of the device means that the Fermi level of the DSMs can be adjusted by adding electrodes to the perimeters of the periodic structures and externally applying a bias voltage. The substrate has a relative permittivity of 3.1, a loss tangent of 0.02, and a thickness of ds = 5 µm. DSMs have a thickness of ts = 2 µm. Scattering and charge transfer from the substrate may have an effect on the properties of the DSMs, but only for very thin layers and contact surfaces. The 2 µm thickness of our DSMs suggests that bulk conductivity would be mostly unaffected. Numerical simulations were performed with CST Microwave Studio. In the simulations, periodic boundary conditions were applied in x and y directions, and an absorbing boundary condition was applied in the z direction. The parameters of the unit cell are as follows: P = 80 µm, L = 68 µm, k = 72 µm, gs = 28 µm, w = 8 µm.
Fig. 1. Geometry of the unit cell of the chiral metamaterial (a) top layer (b) bottom layer (c) profile view (d) isometric view where substrate is shown semi-transparent.
The dynamic conductivity of the DSMs can be written as follows [41]
To study the effects of the DSMs on asymmetric transmission it is necessary to quantify the complex wave amplitudes. The complex amplitudes of the incident waves and transmitted waves can be related by the matrix formalism [43]:
3. Results and discussion
In the simulation, we assume the initial Fermi energy of DSMs is 90 meV. Figures 2(a) and 2(b) show, respectively, the simulated results of the four transmission matrix elements of a linearly polarized wave propagating in forward (+z) and backward (−z) directions. We can see that the co-polarization transmission Txx and Tyy remain equal and do not depend on propagation direction, while the cross-polarization components Tyx and Txy are different across the whole frequency range and swap behavior depending on propagation direction. This asymmetry should lead to a strong AT effect. Figure 2(c) shows the total transmitted intensities (|tx|2 = |Txx|2 + |Tyx|2, |ty|2 = |Tyy|2 + |Txy|2) [46] of the x-polarized and the y-polarized waves propagating in the backward (−z) direction. In this notation, |tx|2 calculates the total intensity of the transmitted wave, normalized to the incident intensity, when an x-polarized wave is incident. This total intensity includes a contribution from both the x- and y-polarized transmitted waves. Likewise, |ty|2 represents the normalized total transmission intensity, including both x- and y-components, of the transmitted wave when a y-polarized wave is incident. It is seen that |tx|2 reaches a maximum value of 0.63, while |ty|2 remains below 0.26 from 1.33 to 1.83 THz. This result indicates that a relatively large portion of incident energy either remains x-polarized or is converted to the x-polarization during transmission in the backward (–z) direction, while a relatively small portion remains or is converted to the y-polarization. For propagation in the + z direction the same behavior is observed except that more energy exits the material in the y-polarization. Figure 2(d) adds more information by showing the polarization conversion ratios (PCRs) for –z propagating waves, where PCRx = Tyx2 / (Txx2+Tyx2), PCRy =Txy2 / (Tyy2+Txy2) [47], and PCRx indicates the ratio of wave power converted to the y-polarization versus the total wave power exiting the material for x-polarized incident waves. PCRx exhibits two peaks of 0.71 and 0.81 at 1.39 and 1.67 THz, respectively, and remains above 0.60 from 1.33 to 1.74 THz. As such the structure can provide relatively high polarization conversion efficiency over a broad bandwidth. PCRy exhibits two much smaller maxima of 0.14 and 0.36 at 1.4 and 1.67 THz, respectively, which also highlights the asymmetric behavior.
Fig. 2. The transmission coefficients when linearly polarized waves propagate in the backward (-z) (a) and forward (+z) (b) directions. The total transmission (c) and polarization conversion ratio (PCR) (d) when the linearly polarized incident waves with either x (red) or y (blue) polarization propagate in the backward (-z) direction.
According to Eq. (5), we calculated the asymmetric transmission parameter of linearly polarized and circularly polarized incident waves, which are shown in Fig. 3(a). The linear AT parameter ($\Delta _{lin}^x$) reaches two peaks of 0.37 and 0.38 at frequencies of 1.38 THz and 1.56 THz, while the circular AT parameter is nearly zero across the whole frequency range. This illustrates how the structure forbids asymmetric power transmission for circularly polarized waves. A perfect AT material would have |$\Delta _{lin}^x$| = 1, which indicates that a stronger polarization conversion mechanism could further enhance AT. We note that the two curves of $\Delta _{lin}^x$ and $\Delta _{lin}^y$ are exactly opposite to each other, meaning both linear polarizations exhibit the same magnitude and character of AT, although of a different sign, another consequence of the structural symmetry.
Fig. 3. (a) The asymmetric transmissions of linearly polarized and circularly polarized waves. (b) Transmission coefficients of circularly polarized wave propagation in the backward (-z) direction. Polarization ellipse of the transmitted wave at (c) 1.38 THz and (d) 1.56 THz for x-polarized wave propagation in the backward (-z) direction. Polarization ellipse of the transmitted wave at (e) 1.38 THz and (f) 1.56 THz for x-polarized wave propagation in the forward (+z) direction.
Figure 3(b) shows the simulated co-polarization transmission coefficients T++ (RCP) and T−− (LCP), and cross-polarization coefficients T+− and T−+. It is apparent that the asymmetric behavior for circularly polarized waves is opposite that of linearly polarized waves. The co-polarized transmission responses for RCP and LCP split into two separate curves from 1.0 to 1.9 THz, while the cross-polarized terms are identical across all frequencies. Given the exclusive role of cross-polarized terms in Δ, it is clear that the structure cannot realize the AT effect when pure circularly polarized waves are incident upon it.
Figures 3(c)–3(f) show the polarization state of the transmitted wave in the backward (–z) and forward (+z) directions for an x-polarized incident wave, at 1.38 and 1.56 THz respectively. In Figs. 3(c) and 3(e), the polarization azimuth of the –z transmitted wave is rotated by −61.8° (‘–’represents clockwise rotation) and for the + z transmitted wave it is rotated + 22.1° (‘+’represents counterclockwise rotation). The polarization azimuth calculation formula is defined as: $\theta = \frac{1}{2}\arctan [{2|{{T_\parallel }} ||{{T_ \bot }} |\cos \varphi /({{|{{T_\parallel }} |}^2} - {{|{{T_ \bot }} |}^2})} ]$. The variables ${T_\parallel }$ and ${T_ \bot }$ are the co-polarization and cross-polarization components, respectively, of the transmitted wave, and φ is the phase difference between the co-polarization and cross- polarization components. At 1.56 THz in Figs. 3(d) and 3(f), the same azimuth rotations are –56.9° and + 7.86°. Also in Figs. 3(c) and 3(e), the transmitted wave acquires an ellipticity angle of 20.1° and 1.6° for –z and + z propagation, respectively. Similarly in Figs. 3(d) and 3(f), the respective ellipticity angles are 14.7° and 23.3°. These results clearly illustrate the asymmetric behavior but also reveal how linearly polarized light changes into an elliptically polarized state as it passes through the chiral structure. The much larger azimuth rotations for –z propagating waves reveal how asymmetric polarization conversion is at the root of AT for this material.
To study the influence of the chiral structure’s geometry on asymmetric transmission, we performed simulations with varying parameters, one at a time: substrate thickness ds, gap width gs, arm width w, and double-T resonator short arm spacing L. These results are shown in Fig. 4 and reveal that some parameters have surprisingly large effects on the bandwidth or asymmetric behavior. As shown in Fig. 4(a), varying the substrate thickness ds from 3 to 5µm affects both bandwidth and magnitude of asymmetric transmission, where larger thickness is better for strong asymmetry, but at the cost of bandwidth. Interestingly, Fig. 4(b) shows that decreasing the gap size, gs, improves the bandwidth but has very little effect on asymmetric transmission magnitude over the passband. From Fig. 4(c), increasing the short arm spacing L increases both bandwidth and asymmetric behavior. Finally, Fig. 4(d) shows that the arm width parameter w has relatively little effect on either bandwidth or asymmetric behavior. Based on these results, we selected the optimized parameter values, described in Section 2.
Fig. 4. Influence of different structural parameters on the asymmetric transmission. (a) the substrate thickness ds (b) the gap width gs (c) the double T resonators short arm spacing L (d) the arm width w.
An examination of the resonance currents reveals why the structure exhibits a wide bandwidth response. As shown in Fig. 5(a), we label the two resonance peaks of the PCR for an x-polarized incident wave as mode 1 and mode 2. Figure 5(b) shows both the current densities and black arrows that depict the general mode structure in top and bottom DSMs corresponding to mode 1 and mode 2, respectively. Figure 5(b) 1 shows how currents in the top and bottom layers oscillate out of phase in mode 1. This anti-parallel nature has the effect of decreasing the energy of the coupled system, thus lowering the mode’s resonance frequency. However, Fig. 5(b) 2 illustrates the second mode, which exhibits parallel currents that oscillate in phase. Here, the parallel currents have the effect of increasing the energy of the coupled system, leading to a higher resonance frequency. Working together, these modes with high and low resonant frequencies extend the overall AT bandwidth.
Fig. 5. (a) PCR of linearly polarized wave, (b) the current densities in the top and bottom layers of the x-polarized wave at 1.389 THz and 1.668 THz, respectively.
Dynamic tuning of the AT behavior occurs when the conductivity of the DSMs increases with increasing Fermi energy. The DSMs begin to behave more like a metallic layer, enhancing the resonance strengths and the resulting polarization conversion. A dramatic change in AT behavior is observed by altering Ef from 20 meV to 100 meV. Figure 6 shows the transmission coefficients and the polarization state of the transmitted waves (x-polarized incident) at different Fermi energy of DSMs. While all of the coefficients evolve with increasing Fermi energy, it is apparent that the co-polarization transmission coefficients |Txx| and |Tyy| evolve almost equivalently, whereas the cross-polarization transmission coefficients |Tyx| and |Txy| quickly diverge in behavior. When the Fermi energy is 40 meV, the cross-polarization coefficients |Tyx| and |Txy| already exhibit significant differences, and antisymmetric transmission can be realized. Interestingly, the cross-polarization constant |Tyx| exhibits a much stronger overall change with Fermi energy, versus |Tyx|, which is always below 0.25 and only suffers a slight blue-shifting behavior. Figure 6(e) shows the polarization state of the transmitted wave. We observe that the rotation angle of the transmitted wave gradually decreases with the increase of the Fermi energy and the polarization goes through various states of ellipticity. Importantly, these results strongly suggest that resonances can be damped and undamped to an extent that makes deep modulation of transmission asymmetry possible. In fact, Fig. 6 shows that this modulation is so deep it can almost enable a discrete device behavior from fully “off” – where there is almost no asymmetry in transmission – to fully “on” – where there is highly asymmetric transmission.
Fig. 6. (a) - (d) The transmission coefficients for x- and y- polarized incident waves propagating in the backward (-z) direction. and (e) polarization ellipse of the transmitted wave at 1.38 THz with different Fermi energy for x- polarized incident waves propagating in the backward (-z) direction.
Figures 7(a)–7(d) shows the transmission coefficients |T++|, |T−−|, |T+−| and |T−+| when a circularly polarized wave (both RCP and LCP circularly polarized waves) is incident in the backward (−z) direction. In this case, the cross-polarization transmission coefficients |T+−| and |T−+| are equivalent at all different Fermi energies, whereas the co-polarization coefficients |T++| and |T−| becomes more different as the Fermi energy increases. From Eq. (5), this means asymmetric transmission of the circularly polarized wave is forbidden at any Fermi energy. Figure 7(e) verifies this by plotting the AT parameters $\Delta _{cir}^ + $ and $\Delta _{cir}^ - $ of circularly polarized waves illuminated in forward (+z) and backward (−z) directions. Since these have a value of zero over the entire parameter space, this behavioral feature is independent of Fermi energy tuning, which may prove useful in certain practical applications.
Fig. 7. (a) - (d) The transmission coefficients with different Fermi energy of DSMs of a circularly polarized wave illuminated in the backward (−z) direction (e) The AT parameters of circularly polarized waves with different Fermi energy of DSMs.
The tunability of AT for linearly polarized incident waves is shown in Fig. 8(a). As Ef changes from 20 to 100 meV, the asymmetric transmission changes quite uniformly from almost zero to 0.37 over a wide bandwidth. The AT band also exhibits an overall blue shift during this transition. Figures 8(b) and 8(c) show how the PCR and total transmission terms |tx|2 and |ty|2 vary with the Fermi level, and that they all have a similar blue shift.
Fig. 8. (a) Asymmetric transmission parameters (b) PCR and (c) total transmission with different Fermi energy for x- and y- polarized incident waves.
Finally, to provide some perspective on the capabilities of DSM-based dynamic control, Table 1 compares the work of this paper with the recent graphene-based asymmetric transmission devices. Graphene-based tunable asymmetric transmission devices are mainly focused on the asymmetric transmission of circular polarization conversion, as described in [48–50], and there are few reports on asymmetric transmission devices for linear polarization conversion. Reference [51] proposes an asymmetric transmission of linear polarization conversion in the terahertz band, but it requires the application of an external magnetic field to control the Hall conductivity of graphene, making the entire device bulky and complicated to operate. Reference [52] proposes a three-layer metal-graphene-metal metasurface to achieve asymmetric transmission, and its asymmetric transmission parameter is about 70%. The structure is anisotropic though not chiral. By adjusting the Fermi level of graphene, the frequency shift of the asymmetric transmission band is not realized, however the on-to-off control of asymmetric transmission is realized. Our work can also realize the asymmetric transmission capability of linear polarization conversion in the terahertz range. Further, by changing the Fermi level of DSMs, both on-to-off control of asymmetric transmission and the blue shift of polarization conversion frequency can be realized.
Table 1. Comparison with recent reported graphene-based asymmetric transmission devices
4. Conclusions
In this paper, we describe a chiral metamaterial structure consisting of two double-T structures made from DSMs on both sides of a dielectric layer. This material can produce asymmetric transmission for linearly polarized waves in the THz region where the AT parameter remains >0.35 over a large bandwidth of 1.38-1.63 THz. Simultaneously, the AT parameter for circularly polarized waves is nearly zero across a range of 1-2 THz, and at any Fermi energy, revealing that this material can also be used for symmetric transmission by properly tailoring the incident polarization state. Importantly, by changing the Fermi level of the Dirac semimetals from 20 meV to 100 meV, the linearly-polarized AT effect becomes quite uniformly tunable over a large range, 1.2 to 1.6 THz. This result offers helpful insights in the design strategies of future dynamic AT materials and provides intriguing possibilities for new devices, such as isolators, multiplexers, and switching networks, based on tunable AT. It further provides important perspective on the possibilities and challenges of utilizing DSMs instead of monolayer materials (e.g. graphene) for dynamic terahertz control devices.
Funding
National Natural Science Foundation of China (61775123, 61875106); Key Technology Research and Development Program of Shandong (2019GGX104039, 2019GGX104053); Shandong graduate student tutor guidance ability promotion program project (SDYY17030); National key research and development program of China (2017YFA0701000); the Scholarship Fund of SDUST.
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