Enhancement of Electromagnetically Induced Transparency in Metamaterials Using Long Range Coupling Mediated by a Hyperbolic Material

Near-field coupling is a fundamental physical effect, which plays an important role in the establishment of classical analog of electromagnetically induced transparency (EIT). However, in a normal environment the coupling length between the bright and dark artificial atoms is very short and far less than one wavelength, owing to the exponentially decaying property of near fields. In this work, we report the realization of a long range EIT, by using a hyperbolic metamaterial (HMM) which can convert the near fields into high-k propagating waves to overcome the problem of weak coupling at long distance. Both simulation and experiment show that the coupling length can be enhanced by nearly two orders of magnitude with the aid of a HMM. This long range EIT might be very useful in a variety of applications including sensors, detectors, switch, long-range energy transfer, etc.


Introduction
The hyperbolic metamaterials (HMMs) have attracted considerable interest. These materials can mimic some of the properties of negative-index materials such as negative refraction [1][2][3][4]. It is also possible to tailor such materials by going from elliptic to hyperbolic dispersion relations [5][6][7]. The density of states in the vicinity of HMM can be large [8][9][10] and this has important consequences for spontaneous emission [11,12]. Since HMM can allow very large surface vectors to propagate, unlike metals, these are finding applications in high-resolution imaging [13][14][15][16][17][18], unusual surface wave [19], cavity [20], scattering [21,22], thermal radiation [23][24][25], and giant photoresponsivity [26]. Recently Biehs et al, Cortes and Jacob have discovered a very long range dipole-dipole interaction in a HMM [27][28][29][30]. This is especially significant in the energy transfer from a donor atom to an acceptor atom and can lead to significant entanglement between two atoms, which is important for quantum information tasks. All such studies demonstrate the importance of a HMM in a range of applications. Other interesting possibilities include coupling of macroscopic objects like nanoparticles or fundamental interaction between a dipole and a quadrupole.
We present a theoretical and experimental study of the coupling between a dipole-like bright atom and a quadrupole-like dark atom which is especially relevant in the classical analog of electromagnetically induced transparency (EIT) based on metamaterials. The realization of the EIT which derives the interference-based high-Q resonance inside the stop band has attracted much attention in terms of various applications: slow light, sensors, switch, etc [31][32][33][34][35][36][37]. We note that the EIT in meta systems like three-level system is well known. This requires significant coupling between the bright atom and dark atom modes. Thus the distance between the bright and dark atoms has to be / 20 / 40  , where  is the operation wavelength. For larger distances this coupling is insignificant and the dark atom remains dark when the bright atom is excited. A fundamental question is then how the coupling can be enhanced. One very good possibility is to use the recently discovered long range coupling effects mediated by the HMM. Inspired by this mechanism, we show that with the aid of a HMM, EIT can also be maintained even when the separation between bright and dark atoms is large.
In this work, we will demonstrate the enhancement effect of HMM for the long range EIT by comparing short distance one in a normal material. In a normal material background such as air, EIT realized by a three-level system does not occur when the separation of bright and dark atoms is very large. This is because the iso-frequency contour (IFC) of the normal material is a closed surface and the high-k modes are excluded. However, for a HMM, it supports large k modes because of the open hyperboloid dispersion [38]. The evanescent fields existing in the normal material can be converted to the high-k propagating waves in the HMM. By using epsilon-near-pole (ENP) HMM which has been validated to have a large dipole-dipole interaction [27], we attempt to achieve a long range EIT. Although the long range coupling occurs for any kind of HMM, the ENP-HMM is the most efficient one (see Appendix A for details). Our experimental work demonstrates that an introduced HMM can greatly enhance the interaction distance between a dipole-like bright atom and a quadrupolelike dark atom previously governed by the near-field coupling.
The paper is organized as follows. At first, we consider a general theoretical model of long range EIT that considers the near-field coupling and the converted-far-field coupling between bright and dark atoms, simultaneously. Then, based on microwave transmission-line systems, we perform both simulations and experiments to check the validity of the model and analyze the physical mechanism of the long range EIT. If the model only has near-field coupling term as in the case of a normal medium, the EIT will disappear when the distance between the bright and dark atoms is large. We then put a HMM in the intervening space of the two atoms and show how EIT is restored. In this long range situation, the establishment of EIT mainly comes from the high-k propagating mode because the near-field coupling is very weak. Combining simulation and experimental results, we show that the HMM can enhance the coupling distance by nearly two orders of magnitude in contrast to the situation in a normal environment. This HMM-mediated long range coupling may provide inspiration for the design of new EIT platforms relevant to sensing and detection.

Theoretical model and analysis
Firstly, we establish a physical model of long range EIT in a three-level system, as is schematically shown in Fig. 1. In this model, both near-field coupling between bright and dark atoms contributed by the ordinary mode in a HMM and the converted-far-field coupling contributed by the extraordinary modes are considered. In Fig. 1, the lower red sphere can be directly excited by the incoming wave in S , which could serve as a bright atom. The upper green sphere cannot be directly excited by the incoming wave and it need be excited via the bright atom. Based on the coupled mode theory, the time-harmonic energy amplitudes for two resonance modes can be defined as From Eq. Then, based on microwave transmission lines (TLs), we perform both simulations and experiments to check the validity of the model. The effectiveness of using TLs to mimic artificial atoms has been verified in [40,41]. In fact, in Eq. (1), if there is no coupling term with phase  , that is, only Fig. 1. Schematic of a long range EIT in a three-level system. This general model considers both near-field coupling and converted-far-field coupling between bright and dark atoms.
the near-field coupling term connected with  exists, the model just describes the case of a conventional EIT in a normal medium. We firstly compare the results from the model and those from the simulations in this conventional EIT case. The three-level artificial atom system and the related parameters are given in Fig. 2(a). In this case, the transmittance T of the system can be written asIn Fig.  2(a), the artificial atoms couple to the incident wave by the TL [40, 41]. The narrow split ring resonator (SRR) connected to the TL can be directly excited by the incident wave propagating in the TL and serves as a bright atom. In contrast, the wide SRR on the side of the narrow SRR acts as the dark atom [42], because it is far from the TL and need be excited through the narrow SRR by the near-field coupling between them. Moreover, to avoid detuning, we choose 1 ) substrate with a thickness of 1.6 h  mm. We first simulate the transmittance spectra by the CST software. When the separation between two atoms is small, e.g., 0.2 s  mm, an EIT window around 0.79 GHz appears, as is shown by the purple scattered open dots in Fig. 2(b). However, when s gradually increases, the EIT peak becomes narrower and narrower and at the same time shallower and shallower, as is shown in Fig. 2(b). When s reaches a large value, the EIT disappears and the transmittance spectrum degenerates into that of the single bright atom. In addition to the simulated transmittance (shown by the scattered open dots), the calculated transmittance spectra based on Eq. (3) are also given in Fig. 2 , see Fig. 2 is large and the electric fields are transferred from the bright atom to the dark atom, as is shown in Fig. 2(d). However, for 5 s  mm,  is very small and the electric fields are mainly concentrated at the bright atom, see Fig. 2(e). So, in a normal environment, the EIT no longer occurs for the large separation. However, the coupling distance can be strongly enlarged if we add a HMM between the two atoms. Although both ordinary and extraordinary modes would be excited in the HMM [43], the nearfield coupling contribution from the ordinary modes is negligible when the coupling distance is long, which is similar to the case of a normal material. For a short coupling distance, the near-field coupling contribution from the ordinary modes will play roles in the establishment of EIT. However, similar to [7], in our designed TLs loaded with lumped elements, the excited modes are mainly the transverseelectric (TE) polarized extraordinary modes. The impact of TM-polarized ordinary modes in our TL system can be neglected (see Appendix B for details). The schematic of the HMM and the related anisotropic 2D-circuit model are shown in Fig. 3(a) x direction to ensure a negative y  [44,45]. The dispersion relation of TEpolarized waves in TLs is described by In the long-wavelength limits, if we do not consider the loss, the effective permittivity and permeability of 2D TLs can be written as [7,46] 0 0 0 0 where 0  and 0  are the permittivity and permeability of vacuum, respectively. 0 C and 0 L are the perunit length inductance and capacitance of the TL, respectively. In conventional classical EIT, the dielectric or air background is lossless, while in HMM-mediated long range EIT, the HMM is lossy. So it is very necessary to study the influence of the loss of the HMM on the long range EIT. For the TLbased effective HMM, the loss mainly comes from the dielectric loss of substrate and the Ohm loss of the loaded capacitor in the x direction. Considering the dielectric loss of substrate, the imaginary part of permittivity can be written as 1 2 Re Tan Re Tan Im Based on Eqs. (5) and (6) Fig. 3(c). For comparison, a general hyperbolic dispersion when =1 y   is also given in Fig. 3(c), as are shown by the blue solid line in Fig. 3(c). In addition to the dielectric loss, considering the Ohm loss of loaded capacitors in the x direction, we can calculate the imaginary part of permeability in the y direction as where c R denotes the Ohm loss of capacitors. However, after calculating the dispersion diagram of HMM, we find that Im y  will not introduce the imaginary part of y k (denoted by Im y k ). This means that Im y  will not introduce absorption in the propagating direction and will not influence the long  Fig. 3(c), we see that Im will introduce small values of Im y k , as are shown by the dashed lines. The appearance of Im y k will increase the absorption and decrease the transmission window of the EIT. Besides considering the effect of loss, we need consider the influence of the type of HMM on the long range EIT. Although both type I and type II HMM can convert the near fields into propagating waves, the direction of the converted propagating wave is determined by the designed HMM. For example, in our system, to make the converted-far-field propagate along the y direction, a type I HMM [16] is needed. If the type of HMM is changed, the direction of the converted propagating wave would change and the wave would not interact well with the dark atom. In this case the EIT would be hard to establish.

Simulations and experiments
Now we add the TL-based effective HMM between the bright and dark artificial atoms. The schematic of the structure to realize the long range EIT is shown in Fig. 4. The effective HMM is added between the two SRRs whose geometric parameters are the same as those in Fig. 2(a). The effective HMM contains 4 (6) unit cells in the x ( y ) direction. The parameters of the two atoms remain unchanged except that 2 1.2 C  pF. Moreover, for convenience, the dark atom is put near a single TL out of the effective HMM. In this sample, the distance between the two atoms is 60 mm which is much longer than the effective coupling length in a normal environment in Fig. 2(a). In the HMM, the phase of the converted propagating wave is equal to the product of the wavevector in the propagating direction and the distance ( s ). For an ENP-HMM, the dispersions are flat lines and the fields are collimated in the y direction. In this case, the converted-far-field phase  in HMM can be written as where 0 k is the wave vector of free space. From Eq. (8), we find again that only the imaginary part of  will lead to the imaginary part of  and thereby introduce the absorption.
Moreover, in our system, in addition to the converted-far-field phase  induced by the HMM, there is an additional propagating phase   from the finite size of the atoms and the added transmission line for dark atom coupling. In the CST simulation, we put four probes at the center of the bright atom, the left edge and the right edge of the HMM as well as the center of the dark atom, simultaneously. Then, we extract the phase difference from the center of the (1) can be derived as where T is given by Eq. (2). In the model, if we need not consider the contribution of ordinary modes, we will not consider the near-field coupling term connected with  . In the simulations, T and R can be directly extracted. In all cases, the absorbance 1 A R T    . The spectra of R , T and A from the model and from the simulations for s 45, 60 and 75 mm are given in the left column (dashed lines) and the right column (solid lines) of Fig. 5, respectively. When the distance 60 s  mm ( 0   ), an EIT window is still clearly seen. In comparison with Fig. 2(a), the effective coupling distance is boosted by nearly two orders of magnitude. In the case of 60 s For different s ,  will be different and the corresponding spectra will change according to Eq. (2). It is seen from Fig. 5   With the aid of HMM, EIT can be well reestablished when the distance of bright and dark atoms is much larger than the normal near-field coupling distance. The EIT spectra is still seen when the distance between two atoms reaches 60 mm.
To emphasize the role of the HMM, we show a systematic comparison of a system that does show the effect (long-range EIT) with the system that does not show it. Transmission line systems with different values of capacitors are used for the working and non-working systems. In Fig. 6(a), the transmission line with 0.1 C  pF mimics the ENP-HMM. The simulated transmission spectra of this system when the distance 60 s  and 135 mm, respectively, are shown in Fig. 6(b). The EIT windows are clearly seen for two distances. In Fig. 6(c) and 6(d), we numerically show the electric field distributions corresponding to the EIT peaks in the cases of 60 s  and 135 mm, respectively. In two cases, the fields are transferred from the bright atom to the dark atoms, which demonstrate the long-range EIT. For comparison, we only change the value of C from 0.1 pF to 30 pF. As a result, 0.73 y   and the transmission line mimics a normal material with a nearly circular dispersion, as is shown in Fig. 6(e). The transmission spectra of this system when 60 s  and 135 mm, respectively, are given in Fig. 6(f). There are no EIT windows in the opaque region. Moreover, the electric fields corresponding to the transmission dips in two cases of different distance are mainly confined in the bight atom, as are seen in Fig. 6(g) and 6(h). Therefore, the comparison between working and non-working systems in Fig. 6 convincingly shows that, with the aid of HMM, long-range EIT can be realized. To see the formation of the long range EIT clearly, now we experimentally study the change of field pattern from the case of a single bright atom to the case of a molecular with bright and dark atoms. For the bright atom, a Lorenz line shape can be seen from the reflectance spectra in Fig. 7(a). The corresponding energy will be collimated in the y direction in the HMM, as are shown by the field distribution in Fig. 7(b) at the resonance frequency of 0.806 GHz. However, for the molecule with bright and dark atoms, a transparent EIT window (marked by the black dashed line) occurs in the reflectance spectra in Fig. 7(c). At this EIT frequency, the energy is transferred from the bright atom to the dark atom, see the measured electric field patterns in Fig. 7(d). In our experiments, the samples are placed on an automatic translation device which makes it feasible and accurate to probe the field distribution using a near-field scanning measurement. To measure the electric fields, the signal launches from the port one of vector network analyzer (Agilent PNA Network Analyzer N5222A) and another antenna (near-field probe) connecting to the port 2 of analyzer are employed to records the electric field pattern. The length of the rod antenna is 1 mm. It is vertically placed 1mm above the TLs to measure the signals of electric fields of the TLs in the 2D plane. The spatial step of scanning the near field is set to be 1 mm in the x and y directions, respectively. The field amplitudes are normalized according to their respective maximum amplitude. Comparing Fig. 5 in the case of 60 s  mm with Fig. 7(c), we see that on the whole the experimental results agree well with the simulated one. The deviations between the simulations and the experiments originate from the discrepancies of material parameters including the dielectric constant of the FR4 substrate and the value of capacitors between the theoretical data and the real fabricated sample. Furthermore, to see the dependence of EIT on the separation s and compare with Fig. 2(c) in the case of a conventional EIT, we plot Fig. 8 in the case of a HMM-mediated EIT. As is seen from Eq. (2), the transmittance depends on the converted-far-field phase  in the HMM. So at first we give the dependence of  on s . The linear relationship between the real part of  and s is shown by the red solid line in Fig. 8(a), as is dictated by Eq. (8). For comparison, the simulated real part of  by extracting the phase difference from the left edge to the right edge of the HMM is also given in Fig. 8(a), as are shown by the scattered stars. It is seen that the scattered stars are nearly located at the red solid line. The dependence of the imaginary part of on is also given in Fig. 8(a), as are shown by the black open circles. The imaginary part of only increases a little bit when increases from 60 to 135 mm. This is because the imaginary part of is much smaller than the real part of , see Fig. 3(c). As a result, in Fig. 6(b) the EIT window only decreases a little bit when changes from 60 to 135 mm. In Fig. 8

Conclusion
As a summary, the phenomenon of long range EIT observed in our experiments confirms the theoretical prediction that the energy transfer will be enhanced strongly in a HMM environment. This experimental work demonstrates that the HMM can overcome the short distance limitation of near-field coupling in a conventional medium. The HMM-enhanced coupling would have profound influences on the physical processes previously governed by the near-field coupling. Moreover, this long range EIT might be very useful in various applications including slow light, sensors, switch, long-range energy transfer, etc.