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Abstract
Metamaterial analogue of electromagnetically induced absorption (EIA) has promising applications in spectroscopy and sensing. Here we propose an EIA metamaterial based on hybrid metal/dielectric structures, which are composed of a metallic wire and a dielectric block, and investigate the EIA-like effect by simulations, experiments, and the two-oscillator model. An EIA-like effect emerges in virtue of the near-field coupling between metallic wire and dielectric block, and the dielectric block exhibiting magnetic dipolar resonance makes a major contribution to the resonance absorption. The magnetic flux through the dielectric block engendered by the near filed of the metallic wire determines the coupling between dielectric block and metallic wire. With the variation of the separation between dielectric block and metallic wire, the EIA-like effect is preserved and does not convert into the EIT-like effect although the coupling and consequently the absorbance are altered. Based on the two-oscillator model, the absorption spectrum of the EIA metamaterial is quantitatively analyzed and the parameters of the oscillator system are retrieved.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a quantum interference effect that reduces light absorption over a narrow spectral region in an atomic system [1]. Due to the destructive inference between excitation pathways, EIT is accompanied by important consequences such as dramatically reducing the group velocity [2] and enhancing nonlinear interactions [3]. So far, most of the proposed techniques to achieve EIT require rigorous experimental conditions, severely hampering the applications of EIT. However, it is demonstrated that the EIT-like effect can occur in electric circuits [4], classical resonator systems [5,6] and metamaterials [7–24]. To achieve miniaturized and versatile metamaterial devices based on EIT-like effect for practical applications, tunable EIT-like effect of metamaterials is extensively researched [25–33]. Contrary to EIT, electromagnetically induced absorption (EIA) originates from the constructive interference of different excitation pathways in an atomic system, leading to a narrow peak in the broad absorption spectrum [34]. Corresponding to the metamaterial analogue of EIT, the EIA-like effect can also occur in metamaterials. In EIT metamaterials, by decreasing the dissipative loss of the bright resonator, increasing the dissipative loss of the dark resonator or decreasing the coupling strength between the bright and dark resonators, the transition from EIT to EIA can be observed [35]. Moreover, the EIA-like effect can be obtained by manipulating the coupling phase between the bright and dark resonators [36–39]. Via making an analogy to the atomic physics concept of EIA in four-level atomic system, three-resonator systems are proposed to mimic the EIA-like effect [40–42]. Besides, the active control [43–45] as well as applications [46] of EIA-like effect is also researched.
A simple and effective metamaterial design for EIA-like effect remains the coupled bright and dark resonator system. However, if the single-material system is adopted, the separation between the bright and dark resonators, which is required to introduce the coupling phase, will increase the thickness of EIA metamaterials and do not adapt to the integrated metamaterial devices. In this work, according to the transition mechanism from EIT to EIA, we select to increase the dissipative loss of dark resonator to achieve EIA-like effect. To meet the requirement of EIA-like effect for the dissipative loss of the bright and dark resonators, we exploit the difference of dissipative loss between metal and dielectric, and propose a hybrid metal/dielectric metamaterial to achieve EIA-like effect. Different from previous EIA metamaterials, our design of EIA metamaterials gets rid of the limitation on the separation between the bright and dark resonators, and has great potential in developing planar EIA metamaterials. Via simulations and experiments, the absorption behavior of the hybrid EIA metamaterial is investigated and the influence of the coupling between metallic wire and dielectric block on the EIA-like effect is examined. To interpret the origin of the resonance absorption in the hybrid EIA metamaterial, the magnetic dipole moment of the dielectric block is calculated. Moreover, the magnetic flux through the dielectric block is also calculated to explain the variation of the coupling with the separation between metallic wire and dielectric block. Finally, the absorption behavior of the hybrid EIA metamaterial is quantitatively analyzed with the two-oscillator model and the parameters of the oscillator system are retrieved by fitting method.
2. Structure design
Considering the requirement of the EIA-like effect for the dissipative loss factor of the bright and dark resonators, we select metal and dielectric as the constituent materials of the bright and dark resonators, respectively. Figure 1 depicts the hybrid EIA metamaterial composed of an I-shape metallic wire and a square dielectric block, which are located on the front and back of substrate, respectively. The metallic wire can be directly excited by the incident field to show an electric dipolar resonance and is consequently a bright resonator; due to the large radiative loss, the electric dipolar resonance of metallic wire exhibits lower $Q$ value. However, the dielectric block is fixed at the center of substrate and can only be excited by the near field of metallic wire to exhibit magnetic dipolar resonance, so it is a dark resonator; the $Q$ value of the magnetic dipolar resonance of dielectric block, which is free from radiative loss, is determined by the dielectric loss of ZrO$_2$-doped CaTiO$_3$ (1 wt.%). The I-shape metallic wire was fabricated with the standard PCB technology and the ZrO$_2$-doped CaTiO$_3$ ceramic is prepared with conventional solid-state method, which is described in detail in Ref. [47]. After calibrating the vector network analyzer (VNA) with the Thru-Reflection-Line (TRL) method, the hybrid metal/dielectric structure is placed inside a standard rectangular waveguide of WR-90 with cross section of 22.86 mm $\times$ 10.16 mm and the scattering parameters are automatically acquired with the VNA (AV3629D). All numerical calculations are carried out through full wave simulations and the boundaries perpendicular to $x$ and $y$ axes are set to PEC.
Fig. 1. Schematic (a), front view (b) and back view (c) of the hybrid EIA metamaterial. The white dashed frame around the cyan square indicates that the dielectric block is located on the back of substrate. The metallic wire is copper with a conductivity of $5.8 \times 10^7$ S/m; the dielectric block is ZrO$_2$-doped CaTiO$_3$ (1 wt.%) with permittivity $\varepsilon =123$ and loss tangent $\tan \delta =0.001$; the substrate is Teflon with permittivity $\varepsilon =2.65$ and loss tangent $\tan \delta =0.001$. The geometrical parameters of metallic wire and dielectric block are as follows: $l_1=7.4$ mm, $l_2=3.0$ mm, $w=0.5$ mm, $a=3.5$ mm; the depth of metallic wire is 0.035 mm and the thickness of dielectric block is 1.0 mm. The dimensions of substrate are 22.86 mm $\times$ 10.16 mm $\times$ 2 mm, which is in accordance with the cross section of the standard rectangular waveguide of WR-90. The dielectric block is fixed at the center of substrate and the metallic wire is moved along $y$ direction to alter the coupling between metallic wire and dielectric block.
3. Results and discussion
We first investigate the effect of asymmetry of the EIA metamaterial on the scattering parameters. When the metallic wire is located at the center of the substrate, i.e. $s=0$ mm, the net magnetic flux through the dielectric block vanishes because of the symmetry of the EIA metamaterial and thus the dielectric block can not be excited. As a result, the EIA metamaterial behaves as if the dielectric block does not exist. The transmission spectrum shown in Fig. 2(a) has a dip at 8.87 GHz, indicating the electric dipolar resonance of the symmetric EIA metamaterial; moreover, the absorption spectrum, which is obtained with $A=1-|S_{21}|^2-|S_{11}|^2$, demonstrates that the absorption of the symmetric EIA metamaterial is negligible. After the metallic wire is moved along $y$ direction, for example, $s=0.2$ mm, the EIA metamaterial is symmetry-broken and the dielectric block is excited to exhibit magnetic dipolar resonance due to the nonzero magnetic flux generated by the near filed of metallic wire. As shown in Fig. 2(c), the transmission spectrum has a passband and the reflection spectrum has a dip at 8.91 GHz owing to the magnetic dipolar resonance of dielectric block; in virtue of the dielectric loss of the resonant dielectric block, a sharp absorption peak with absorbance of 0.40 occurs at 8.91 GHz in the dim background, and the $Q$ value of absorption peak is up to 441. The experimental results are in good agreement with the numerical results. As a matter of fact, if both the metallic wire and the dielectric block are placed asymmetrically on the same side of the substrate, the EIA effect remains as long as the geometrical parameters of the metallic wire and dielectric block match with each other. For instance, when the geometrical parameters $a$, $s$ and the thickness of dielectric block change from 3.5 mm, 0.2 mm and 1.0 mm to 5.5 mm, 0.4 mm and 0.5 mm, respectively, a similar EIA effect to Fig. 2(a) can be observed.
Fig. 2. Transmission, reflection and absorption spectra of the EIA metamaterial with $s=0$ mm (a, b) and $s=0.2$ mm (c, d). (a, c) and (b, d) are acquired from simulations and experiments, respectively.
In the EIA metamaterial, all the constituent materials are lossy, so all the components, i.e. dielectric block, metallic wire and Teflon substrate, contribute to the absorption of EIA metamaterial. By integrating the loss density, the absorption spectra of dielectric block, metallic wire and Teflon substrate are calculated and shown in Fig. 3. It is seen that the absorbance of dielectric block is about 38 times larger than that of metallic wire and Teflon substrate at 8.98 GHz and therefore the dielectric block is the major contributor to the resonance absorption, indicating that the absorption of EIA metamaterial mostly stems from the magnetic dipolar resonance of dielectric block. The absorption dip of metallic wire arises from the suppression of electric dipolar response by dielectric block, and the absorption peak of Teflon substrate is induced by the magnetic dipolar resonance of dielectric block.
Fig. 3. Calculated absorption spectra of dielectric block $A_1$ (a), metallic wire $A_2$ (b) and Teflon substrate $A_3$ (c) in the EIA metamaterial with $s=0.2$ mm.
The coupling between dielectric block and metallic wire plays a key role in the absorption of EIA metamaterial. We move the metallic wire along $y$ direction to alter the coupling and investigate the influence of coupling on the EIA-like effect. As shown in Fig. 4, with the increment of separation $s$, the absorption spectrum of EIA metamaterial preserves the EIA-like lineshape, indicating that the proposed hybrid metamaterial has a stable EIA-like effect, but the peak absorbance varies non-monotonously with the extremum reached at $s=3.0$ mm. When the separation $s$ is less than 3.0 mm, the peak absorbance decreases with the increasing separation $s$; nevertheless, for the separation $s$ larger than 3.0 mm, the larger the separations $s$ is, the higher the peak absorbance grows. The peak absorbance reaches the minimum 0.13 and maximum 0.56 at $s=3.0$ and $s=8.0$ mm, respectively. The experimental results match well with the numerical results.
Fig. 4. Absorption spectra of the EIA metamaterial with the separation $s$ ranging from 0.2 mm to 8.0 mm obtained with simulations (a-d) and experiments (e-h). (i) Relation between peak absorbance and separation $s$.
As mentioned previously, the absorption of EIA metamaterial originates from the magnetic dipolar resonance of dielectric block. To understand the relation between the absorption and the magnetic dipolar resonance, the magnetic dipole moment of dielectric block is calculated according to the formula $\vec {m}=\frac {1}{2} \int \vec {r}\times \vec {j}_p dV=\frac {-i\omega (\varepsilon -\varepsilon _0)}{2}\int \vec {r}\times \vec {E} dV$ by numerical integration. Moreover, the magnetic flux through the dielectric block is also calculated to demonstrate the effect of coupling on the magnetic dipolar resonance of dielectric block and subsequently the absorption of EIA metamaterial. Figures 5(a)–5(d) demonstrate the magnitude of magnetic dipole moment $m$ of the dielectric block in EIA metamaterial. It is seen that the variation of $m$-spectrum with separation $s$ is non-monotonous. The peak value of $m$ first decreases and then increases as the separation $s$ increases. The peak value of $m$ reaches the minimum $0.86\times 10^{-5}$ A$\cdot$m$^2$ and maximum $1.75\times 10^{-5}$ A$\cdot$m$^2$ at $s=3.0$ and $s=8.0$ mm, respectively. By comparing Figs. 5(a)–5(d) with Figs. 4(a)–4(d), it is demonstrated that the variation of $m$-spectrum with separation $s$ is analogous to that of absorption spectrum, corroborating that the absorption of EIA metamaterial mostly stems from the magnetic dipolar resonance of dielectric block. As a matter of fact, it will be seen later that the absorbance $A$ is proportional to the square of magnetic dipole moment $m$. Figures 5(e)–5(h) illustrate the magnitude of magnetic flux ${\varPhi }$ through the dielectric block generated by metallic wire. Since the magnetic dipolar response of dielectric block is excited by the near field of metallic wire via electromagnetic induction, the magnetic flux ${\varPhi }$ determines the coupling between dielectric block and metallic wire. It is indicated that the coupling between dielectric block and metallic wire is enhanced first and then weakened as the separation $s$ increases. After comparing Figs. 5(a)–5(d) with Figs. 5(e)–5(h), it is concluded that enhanced coupling results in weak magnetic resonance.
Fig. 5. Magnetic dipole moment $m$ of the dielectric block in EIA metamaterial (a-d) and magnetic flux ${\varPhi }$ through the dielectric block generated by metallic wire (e-h) with the separation $s$ ranging from 0.2 mm to 8.0 mm. When calculating $m$ and ${\varPhi }$, the incident power at port is 1 W. The units of $m$ and ${\varPhi }$ are $\times 10^{-5}$ A$\cdot$m$^2$ and $\times 10^{-10}$ Wb, respectively.
To give a quantitative description of the EIA metamaterial, the two-oscillator model [9] is utilized to reproduce the absorption spectrum and retrieve the coupling coefficient between metallic wire and dielectric block. The metallic wire is represented by oscillator 1, which is driven by incident field $E$ and is thus a bright oscillator; the dielectric block is represented by oscillator 2, which can be excited only through the coupling between the two oscillators and is therefore a dark oscillator. The dynamic equation of resonance mode $q_1$ and $q_2$, which represent the electric dipole moment of metallic wire and the magnetic dipole moment of dielectric block, respectively, can be written as
For the case of time-harmonic excitation, i.e. $e^{-i\omega t}$, the stationary solution of Eq. (1) can be expressed as
Since the absorption of EIA metamaterial mainly arises from the dielectric block, the absorption spectrum can be approximately expressed as [35]
By fitting the simulated absorption spectrum of the EIA metamaterial with $s=0.2$ mm according to Eq. (3), the parameters of the oscillator system can be acquired as follows: $\omega _1=2\pi \times 8.73$ GHz, $\omega _2=2\pi \times 8.91$ GHz, $\gamma _1=2\pi \times 3.67$ GHz, $\gamma _2=2\pi \times 0.00916$ GHz, $\kappa =2\pi \times 0.108$ GHz.
It is manifest that the fitting resonance frequencies $\omega _1$ and $\omega _2$ are consistent with the transmission dip of single metallic wire and the transmission peak of EIA metamaterial, respectively, as shown in Figs. 2(a) and 2(c). The damping factor $\gamma _1$ of metallic wire is about 400 times larger than the damping factor $\gamma _2$ of dielectric block, suggesting a large radiative loss of metallic wire. Furthermore, the coupling coefficient $\kappa$ is much less than the resonance frequencies $\omega _1$ and $\omega _2$, indicating a weak coupling between metallic wire and dielectric block. The fitting absorption spectrum as well as simulated absorption spectrum of the EIA metamaterial with $s=0.2$ mm is shown in Fig. 6(a). It is apparent that the two-oscillator model reproduces the simulated absorption spectrum of the EIA metamaterial, especially around the absorption peak, demonstrating that the interaction between the EIA metamaterial and the incident wave can be well described by the two-oscillator model. Figure 6(b) illustrates the influence of separation $s$ on the coupling coefficient $\kappa$ and peak value of $m$. It is seen that the coupling coefficient $\kappa$ first increases and then decreases as the separation $s$ increases, and reaches the maximum $2\pi \times 0.646$ GHz at $s=3.0$ mm. The variation of coupling coefficient $\kappa$ with separation $s$ is similar to that of magnetic flux ${\varPhi }$ shown in Figs. 5(e)–5(h), confirming the foregoing statement that the magnetic flux determines the coupling between dielectric block and metallic wire. By comparing the coupling coefficient $\kappa$ and peak value of $m$ as a function of separation $s$, it is verified again that enhanced coupling results in weak magnetic resonance.
Fig. 6. (a) Reproduced absorption spectrum of the EIA metamaterial with $s=0.2\,\rm{mm}$ by the two-oscillator model. For comparison, the simulated absorption spectrum is also plotted in (a). (b) Coupling coefficient $\kappa$ and peak value of $m$ as a function of separation $s$. The units of $\kappa$ and $m$$_{\rm{peak}}$ are $\times 2\pi \,{\rm{GHz}}$ and $\times 10^{-5}\,\rm{A} \cdot \rm{m}^{2}$, respectively.
4. Conclusion
In summary, we present a hybrid metal/dielectric EIA metamaterial composed of a metallic wire and a dielectric block and investigate the EIA-like effect by the simulations, experiments, and two-oscillator model. An EIA-like effect appears as a result of the weak coupling between metallic wire and dielectric block and the resulting magnetic dipole resonance of dielectric block. The coupling between metallic wire and dielectric block is determined by the magnetic flux through dielectric block. As the separation $s$ varies, the coupling is altered but the EIA-like effect is preserved and does not convert into the EIT-like effect. The two-oscillator model fits the EIA metamaterial and the parameters of the oscillator system are retrieved by fitting the absorption spectrum. The results in this work may be useful in application areas, such as spectroscopy and sensing.
Funding
Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20170817162221169); National Natural Science Foundation of China (11372248, 11674266, 61505164, 61771402); Natural Science Basic Research Plan in Shaanxi Province of China (2020JM-145, 2018JM6024); Hong Kong Scholars Program (XJ2017006); Fundamental Research Funds for the Central Universities (3102017zy033, 3102018jgc008).
Disclosures
The authors declare no conflicts of interest.
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