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Abstract
In this work, we demonstrate an analogue of electromagnetically induced reflection (EIR) effect with hybrid structure consisting of a silica (SiO2) square array layer embedded in graphene-dielectric-Au film constructed F-P cavity. It is shown that the SiO2 square array and F-P cavity create transverse waveguide with high quality factor (Q-factor) and longitudinal F-P modes, and their destructive interference effectively forms the EIR-like effect, which benefits for obtaining high group delay. In addition, the C4 symmetric structure ensures the polarization-independent for this EIR-like effect. With high Q-factor at the reflection window, the ultra-high group delay as high as 245 ps can be obtained. This structure will be useful to develop the EIT-like devices with excellent performance such as high group delay, polarization and incident insensitivity, and environmental stability.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Traditional electromagnetic induced transparency (EIT) is a quantum phenomenon, which is formed by the interference between two different excitation paths in the three-level atomic system [1,2]. This interference forms a transparent window with strong dispersion which can be supposed to be used for the slow-light manipulation and the enhancement of nonlinear effects [3–5]. However, the harsh experimental conditions of EIT limit its further extensive applications. Recently, EIT-like device based on metamaterials (MM) has been proposed and experimentally achieved, showing great potential in slow-light devices and relative fields [6–11]. Metamaterials with EIT-like effect usually exhibits EIT phenomena by forming destructive interference through near-field coupling between bright and dark resonance modes [7,12–14].
Group delay is one of the key parameters in digital communication system and an important performance description parameter in EIT-like metamaterials. Traditionally, by adjusting the excitation path or the distance of near-field coupling or conductive coupling between resonators, the group delay of THz pulse is usually at a time scale below 30 ps as previously reported [15–21]. Most of these EIT-like devices consist of a patterned conductive material layer and a low loss transparent dielectric substrate. Zhao et al. [22,23] reported two metamaterials with high group delay successively. One is a bilayer metasurface with an optimized group delay of 40.4 ps, and the other is a spoof localized surface plasmon circular cavity with an optimized group delay of 46 ps. The group delay of 46 ps is currently the largest group delay value obtained in metamaterials in THz region [23]. In fact, according to the coupled-mode theory, Qiang Li et al. designed a high quality factor (Q-factor) and strong-coupling model to obtain a group delay of hundreds of ps in the infrared region [24]. This means that there is still a lot of room to be explored in the group delay phenomena.
Due to the fact that the ohmic loss of dielectric metasurface resonances is usually much smaller than that of the surface plasmon resonance mode, the dielectric materials are used to replace metals, thereby obtaining high Q-factor EIT-like effect metamaterials in visible-infrared band [6,25–28]. In THz band, the materials that can achieve surface plasmon resonance modes are typically graphene and Dirac semimetals [29], which have high losses as well. Usually, the low-loss dielectric materials such as SiO2 has a low refractive index and is difficult to form a binding mode for terahertz waves; and thus, it is difficult to design EIT-like dielectric metamaterials with high group delay. Moreover, the dielectric materials with high refractive index have high loss in the terahertz band, and the group delay of the designed EIT metamaterial is relatively low [7,29]. High group delay has great application prospects in the field of slow-light devices. For example, in optical buffer devices, a higher group delay is more conducive to improve the storage density of optical buffer devices [30]. Fuli Zhang [7] and Lei Han [29] have designed special polarization-insensitive terahertz EIT devices, however the obtained group delay values still need further improvement. In addition, for most EIT-like metamaterials, it is necessary to have both bright and dark modes [13,14,31]. EIT-like effect occurs when the frequencies of the two modes are close to each other. In the EIT-like metamaterials, the oscillate directions of the two resonance modes are mainly perpendicular to each other, and thus a non-C4 symmetry geometric structure is usually needed, which then leads to a polarization-direction sensitive effect. Therefore, in practice, these EIT-like devices are strongly dependent on the polarization direction of the incident light, which increases the difficulty for practical applications. Therefore, a terahertz EIT-like metamaterial with polarization-insensitive and high group delay is still highly required to obtain high-performance optical devices such as optical buffer devices.
As a common device, F-P cavity is widely used in laser devices and THz absorbers [32–34]. And metamaterials combining the dielectric metamaterial and the F-P cavity has also been reported in visible-infrared region [6,35]. As a common resonant cavity, the F-P cavity can form destructive interference resonance modes in the cavity [36,37]. In principle, this longitudinal resonance mode perpendicular to any transverse resonant mode can be utilized to achieve the EIT-like effect which is insensitive to incident light polarization. Therefore, the combination of F-P cavity with SiO2 metamaterial layer is beneficial to obtain polarization independent EIT-like device in THz region.
In this work, we design a device to achieve EIR-like effects by coupling the longitudinal and transverse resonance modes. The device consists of a graphene-dielectric spacer-Au film F-P cavity structure with a SiO2 square array embedded in it. The SiO2 square array is utilized to form a narrow-band transverse waveguide mode coupling with the longitudinal resonance from the F-P cavity, achieving an EIR-like phenomenon. The C4 symmetric structure is insensitive to the polarization, and the reflection peak position remains unchanged in a wide range of incident angles. The group delay increases with the Q-factor increases. Considering the reflectivity above 55%, the group delay reaches 245 ps. These properties show great potentials in the development of new slow-light photonic components and photonic integrated chip-level buffer devices.
2. Structural Design
Figure 1(a) shows schematic diagram of the designed structure, with two wanted modes in it. And Fig. 1(b) shows the expected absorption modes for obtaining EIT effect. The structure is an F-P cavity hybrid metamaterial with graphene-medium-Au film sandwich structure in which a SiO2 square array is embedded. Among them, the Au film is modeled as a reflective layer with a conductivity of 4×107 S/m, and a thickness of t = 2 µm [38]. The SiO2 (nsilica = 2 [39]) square array is located on the surface of Au, with the square thickness of h, the side length of a, and the periodicity of D. The intermediate dielectric spacer layer is filled with TOPAS (a kind of cyclic olefin copolymer commodity with a refractive index of nTOPAS = 1.53 [40]) with a thickness of H. The upper surface is a single-layer planar graphene structure. In terahertz band and at room temperature of T = 300 K, the surface conductivity of graphene, σS (ω), can be expressed by Drude-like form, which is [41,42]:
The narrowband of the waveguide mode in the designed EIT-like system has a high Q-factor which is beneficial for obtaining high group delay [24]. In the structure shown in Fig. 1(a), the designed the metamaterial layer is composed of a silica square array which can produce a high group delay narrowband waveguide mode, and the F-P cavity structure formed after the introduction of the graphene layer can produce longitudinal interference cancellation F-P mode with broadband absorption. The EIR-like effect in this structure is produced by the coupling between F-P mode and waveguide mode.
Modeling and numerical simulation of the proposed sample are performed with Finite Difference Time Domain (FDTD) method to verify the design above. Periodic boundary conditions are applied in x and y directions, and PML absorbing boundary condition is applied in the z directions. Since graphene is a single-layer carbon atom material, the graphene layer structure is simulated as an equivalent two-dimensional thicknessless surface impedance layer. We assume that the initial chemical potential of graphene μc = 0.5 eV, the relaxation time τ = 0.1 ps, and the temperature T is fixed at 300 K. THz plane waves are along the z direction normally illuminated on the device.
3. Results and discussions
In order to show the high Q-factor property and also the high group delay for the transverse waveguide mode provided by the structure, we firstly show case of the structure without graphene [the insert of Fig. 2(a)]. For the specific parameters of h = 20 µm, a = 50 µm, D = 60 µm and H = 85 µm, a narrow dip with a Q-factor as high as 825 at 2.883 THz is obtained. Hereby, Q-factor is calculated by the formula: Q = fc/FWHM (fc is the central frequency, FWHM is the full width at half maxima). This means that this transverse resonant mode acts as a metastable state in the EIT-like effect system, and such a high Q value can be conducive to obtain a high group delay [26]. Note that the peak position of the transverse mode mainly depends on the size of the SiO2 square array, and the size of H does not obviously affect the peak position. We can further calculate the group delay using the relationship τg=-dφ/dω with φ the phase of the reflected light and ω the angular frequency of the electromagnetic wave and obtain a sufficient high value of 200 ps (at 2.883 THz). This could therefore be beneficial to obtain high group delay when another mode with broadband absorption is coupled with it. For this purpose, we add a layer of graphene to the structure in Fig. 2(a) to form the F-P cavity, which can produce the longitudinal interference cancellation mode with broadband absorption [40]. The longitudinal resonance mode hereby is provided by the structure of F-P cavity, which is strongly dependent on the spacer thickness H, and so it is called F-P mode. The F-P mode is then discussed based on the structure without SiO2 square array (the insert in Fig. 2(b)). And the curve in Fig. 2(b) shows corresponding reflection spectra, which exhibits a broadband absorption mode property. Obviously, the F-P mode oscillates vertically, which is perpendicular to the oscillation direction of the waveguide mode. Therefore, the introduction of F-P mode is conducive to couple with the waveguide mode and then obtain the EIR-like effect.
Fig. 2. (a) Reflection spectra and group delay of the structure without graphene layer, with h = 20 µm, a = 50 µm, D = 60 µm, and H = 85 µm. (b) Reflection spectra and group delay of the structure without SiO2 square array with H = 85 µm.
The absorption of F-P cavity mode in the F-P cavity hybrid structure based on graphene-dielectric-Au film can be calculated according to the following formula: A = 1 - Ttran - R, where Ttran is the transmittance and R is the reflectance. However, the metal reflective layer designed here is a perfect electrical conductor, thus the transmittance Ttran = 0. As a result, the absorption efficiency can be simplified to A = 1 - R. Using the Fresnel formula, the absorption efficiency of the structure under normal incidence is derived as [40]:
where ${\xi } + i{\zeta } = [{Re({{{\sigma }_{s}}} )+ iIm({{{\sigma }_{s}}} )} ]{{Z}_0}/{{n}_{1}}$ in which Re(σs) and Im(σs) represent the real and imaginary parts of graphene’s conductivity, Z0 is the free space impedance, n1 is the refractive index of the upper surface of the graphene medium (n1 = 1in this work). And Φ = 2πn0Hf/c-mπ, in which n0 is the refractive index of the spacer dielectric (n0 = nTOPAS = 1.53), m is a natural number, c is the speed of the electromagnetic wave in vacuum, and H is the thickness of the spacer. And the F-P mode appears when let ${{A}_{({f} )}}^{\prime} = 0$ and ${A_{(f )}}^{\prime\prime} < 0$. Let A = 1, the Eq. (3) can be derived toFor the F-P mode, when ζ= 0 (i.e. Im(σs) 0), the Eq. (5) is simplified to the traditional F-P cavity interference formula,
where m is a natural number representing the order of the F-P mode. In order to determine the value of m conducive to the EIT effect, we first calculate using formula (6) to obtain H value to investigate the properties of the total hybrid structure (shown in Fig. 1(a)). When f = 2.883 THz, we obtain H of 17 (m = 0), 51 (m = 1), 85 µm (m = 2), …. Since there is no EIT-like effect appearing for the hybrid structure at H = 51 µm (black curve in Fig. 3(a)), and the EIT-like effect appears at H = 85 µm (red curve in Fig. 3(a)), the order of the F-P mode should set to be m = 2 for further investigation.Fig. 3. (a)The reflection spectra with H varying from 50 to 100 µm. (b) Reflection spectra of the hybrid structure with H =51 (m = 1) (black curve) and 85 µm (m = 2) (red curve). (c) Group delay of the hybrid structure as a function of frequency at H = 85 µm. The parameters of h = 20 µm, a = 50 µm, and D = 60 µm are applied for (a)-(c).
In general, the F-P mode absorption peak follows the absorption law (Eq. (6)) of interference cancellation of F-P cavity. However, the values of imaginary part of graphene’s conductivity Im(σs) and the permittivity of silica square array and TOPAS will affect the absorption peak position of the F-P mode. Therefore, it is necessary to further investigate the influence of H on the EIT-like phenomenon. the other parameters for adjusting the absorption peak position of the F-P absorption mode by changing H to couple with the waveguide mode of the silica square array metamaterial layer are fixed at h = 20 µm, a = 50 µm, and D = 60 µm. We analyze the reflection spectra in the process of H changing from 50 to 100 µm (as shown in Fig. 3(b)). It can be seen that there is a slight redshift mode near 2.883 THz with the increase of H. This means that the resonance mode near 2.883 THz is a transverse waveguide mode produced by the silica square array [6]. When H = 85 µm, the peaks of the two modes overlap and an obvious EIT-like phenomenon is observed in the reflection spectrum (the red curve in Fig. 3(a)). And it shows a distinct EIR-like window peak at 2.883 THz in the reflection spectrum with a high reflection of about 75% is observed at H = 85 µm, and the corresponding Q-factor is 580 (the red curve in Fig. 3(a)). These results confirm that the F-P mode and the waveguide mode can be effectively coupled and achieve EIR-like phenomenon.
As an important index to study the slow light performance of devices, the group delay has also been studied in this work. When H = 85 µm, a = 50 µm, h = 20 µm, and D = 60 µm (corresponding reflection spectrum is in Fig. 3(a)), the calculated group delay is presented in Fig. 3(c), with the highest group delay of 105 ps reached at 2.883 THz. This value is about half of that without graphene case, however still much higher than the reported maximum value of 46 ps caused by spoof localized surface plasmon in the terahertz region [23]. Qiang Li et al. [24] studied the EIT-like phenomenon in waveguides with high Q cavities by means of coupled mode theory, and proved that the coupling of waveguide mode with high Q value is beneficial to form high group delay EIT-like effect. The achieved high group delay in this work is just benefited from the high Q-factor of the waveguide mode caused by silica square array. In addition, group delay for our structure are normal dispersion and slow-light in the two splitting dips as shown in Fig. 3(c), which is different from most previous works with abnormal dispersion and fast-light occur in the two splitting dips [14,18,29,43,44]. This phenomenon is similar to the over-coupled case reported by Qiang Li et al. [24].
The influence of the silica square array’s geometric parameters on the EIT-like effect has also been investigated. Since the array is both the part of waveguide excitation layer and the transmission layer, the periodicity, side length and thickness of the square array can all affect the peak position of the reflection window. Figure 4(a) shows the reflection spectra at h = 15, 20, 25 and 30 µm, respectively, with H = 85 µm, a = 50 µm, and D = 60 µm. The detailed peak position change curve of the reflection window is shown in the insert of Fig. 4(a). It can be seen that both the F-P mode absorption peak and the reflection window positions gradually redshift as h increases. However, the rate of the reflection window redshift is relatively faster than that of the absorption peak of the F-P mode. In addition, the reflectivity increases first, and then decreases, and the highest reflectivity of 75% is achieved at h = 20 µm for the reflection windows. Therefore, h = 20 µm is used in the following studies.
Fig. 4. Reflection spectra of the structures at different (a) h, (b) D, and (c) a, at H = 85 µm. (d) Q-factor and the maximum group delay varying with a.
Figure 4(b) shows the reflection spectra with D varying from 56 to 64 µm. With the increase of D, the reflection window gradually redshifts and presents a nearly linear change characteristic (as shown in the insert of Fig. 4(b)), which is in line with the change trend of the grating excited waveguide absorption peak. On the contrary, the absorption peak of the longitudinal model gradually blueshifts with the increase of D.
Figure 4(c) shows the reflection spectra at a = 40, 45, 50, 55, 59 and 60 µm, respectively, at H = 85 µm, h = 20 µm, and D = 60 µm. It can be seen that the peak positions of the reflection window and the F-P mode both redshift with the increase of a (as show in the insert of Fig. 4(c)), and the reflectivity at the reflection windows decreases. However, the Q-factor and the maximum group delay both increase with the increase of a (as shown in Fig. 4(d)), showing a similar trend. When considering the reflectivity at the reflection windows over 55% at a = 55 µm, the maximum group delay is about 245 ps with a Q-factor of 1494. With a further increasing to 59 µm, there is a dip but not reflection window appears in the resonance notch. That is to say, the EIT-like effect disappears with further increase of the Q-factor [24].
In order to better understand the mechanism of EIR-like effect, we further investigate the field distribution characteristics of the EIR-like effect. Unlike general EIT effect, the longitudinal mode here cannot be distinguished from the electric field distribution in x-y plane. Considering that the resolution of the waveguide mode is along the direction of the electric field, we monitor the electric field distribution in x-z plane. To analyze the mechanism of EIR, we fix H = 85 µm, a = 50 µm, h = 20 µm, and D = 60 µm, and monitor the electric and magnetic field distributions. Figures 5(a)-(c) present the electric field distribution at 2.853 THz (dip1), 2.883 THz (peak) and 2.900 THz (dip2), respectively. E-intensity and vector distribution at 2.853 THz and 2.900 THz both show longitudinal and transverse resonance mode properties. The electric field intensity distribution at the reflection window of 2.883 THz shows a strong local electric field in silica on the upper surface of Au, and electric field vector exhibits an obvious waveguide mode property in the F-P cavity. However, there is no F-P mode electric field distribution property at 2.883 THz, which means an interference cancellation effect appears here. And also, the magnetic field distributions at x-z plane are monitored at 2.853 THz (Fig. 5(d)), 2.883 THz (Fig. 5(e)) and 2.900 THz (Fig. 5(f)), respectively. The magnetic and electric field strength at 2.883 THz is one order of magnitude higher than those at 2.853 THz and 2.900 THz. This indicates that the two modes have an effective interference cancellation effect and result in the F-P mode disappearance at 2.883 THz, which means an EIR-like effect appears at 2.883 THz [26]. Normally, the direct coupling between the F-P mode and the waveguide mode is impossible because their propagations are in perpendicular directions. However, coupling occurs when one mode can be employed as an excitation source for the other one [6]. In this work, the F-P mode resonance direction is along z axis and perpendicular to the polarization direction of the incident light, which means that the F-P mode can act as the incident light to coupling with guide mode of the silica array, and then excite the waveguide mode. Hereby, the longitudinal mode can be employed as an indirect excitation source to excite the transverse waveguide mode, and thus causing the EIT phenomenon.
Fig. 5. The electric field distribution at x-z plane at (a) 2.853 THz (dip1), (b) 2.883 THz (peak), and (c) 2.9 THz (dip2). The magnetic field distribution at y-z plane at (d) 2.853 THz (dip1), (e) 2.883 THz (peak), and (f) 2.9 THz (dip2).
The stability of the electromagnetic-induced reflection effect of the structure to the incident angle is also investigated. Figure 6(a) shows the reflection spectra of TM polarization wave with incident angle varying from 0 to 70 degree. It can be seen that the peak position of the reflection window remains unchanged with different incident angles, which indicates that this structure is insensitive to the incident angle. In addition, we have also studied the influence of incident angle on the group delay and the reflection efficiency of the EIR-like peak position. Figure 6(b) shows the corresponding reflectivity and the maximum group delay values at the EIR-like reflection peak with the incident angle varying from 0 to 70 degree. It can be seen that the maximum group delay value gradually decreases as the incident angle increases. When the incident angle reaches 70 degrees, the time group delay is reduced to 60 ps. In addition, as the incident angle increases from 0 to 50 degrees, the reflectivity gradually decreases from 75% to 60%; and when it increases from 50 to 70 degrees, the reflectivity gradually increases to 76%. In the process of changing the incident angle, the group delay still shows a group delay of 60 ps within the incident angle of 70 degrees, while the reflectivity maintains above 60%.
Fig. 6. (a) Reflection spectra of TM polarization wave with incident angle varying from 0 to 70 degree. (b) Reflectivity and the maximum group delay at the reflection peak with the incident angle varying from 0 to 70 degree. (c) Reflection spectra at various polarization angles ranging from 0 to 90 degree, and at circular polarization. (d) Reflection spectra at various environmental refractive index n varying from 1.0 to 1.4.
Most previously reported EIT metamaterials are polarization sensitive for the needs of bright and dark modes [13,14,44,45]. Here, the F-P mode formed by the destructive interference in F-P cavity is a longitudinal resonance which is perpendicular to any transverse resonance. Therefore, when the resonant frequencies of the two modes are close to each other, it is favorable to form EIT phenomenon. The C4 symmetric structure is thus also used to obtain polarization-insensitive EIT-like effect.
To verify the polarization insensitivity of the designed structure, we investigate the sensitivity of the EIR-like effect to the polarization angle. Figure 6(c) shows the reflection spectra when the polarization angle changes from 0 to 90 degrees. It is seen that, although the polarization angle changes, the reflection spectrum does not change correspondingly. In fact, an incident light of any polarization direction can be split into two components in both x- and y-axis directions. The C4 symmetric structures are polarization insensitive to the incident light as it can be broken into two components of polarization along the x-axis and y-axis, separately. Since there is no further resonance direction requirement for the two modes in this work to cause EIR-like effect, the metamaterial can be designed as any C4 symmetrical structures according to the polarization insensitivity. This is then overcome the polarization sensitivity of the traditional slow-light metamaterials [7].
In practical applications, slow light devices usually need to directly contact with the environment. Therefore, environmental stability is a factor that needs to be considered in the application process. However, most slow light devices based on EIT-like effect show high environmental refractive index response, which is obviously not conducive to the application of slow light devices in environmental conditions. The dielectric metamaterial layer of the designed structure is located inside the device (in other words, isolated from the environment), hence the structure is insensitive to the environmental refractive index. Figure 6(d) shows the reflection spectra of the device under different environmental refractive index conditions. It can be seen that the reflection window of the device remains unchanged during the change of the environmental refractive index from 1.0 to 1.4. Therefore, the designed structure shows good environmental stability and can be well used for slow-light devices under environmental conditions. In addition, the reflectivity slightly increases as the environmental refractive index increases from 1.0 to 1.4. This is mainly because the upper surface of graphene is in direct contact with the environment. It can also be calculated by Fresnel's law that the absorption efficiency of F-P cavity decreases as the environmental refractive index increases [40].
4. Conclusion
In conclusion, we report an EIR-like metamaterial with a silica square array layer being embedded into the F-P cavity. The two coupling modes for producing EIR-like effect in reflection spectra are a longitudinal destructive interference mode produced by the F-P cavity, and a transverse waveguide mode produced by the lattice of the silica square array layer and the Au film, respectively. Due to the high Q-factor at the reflection window, the group delay over 245 ps can be obtained with reflectivity over 55% and a Q-factor of 1494. Furthermore, according to the symmetric configuration, this EIR-like metamaterial possesses independent characteristics on the polarization and incidence angles. In addition, the designed structure is insensitive to environmental refractive index. Considering strict requirement for polarization independence and incident angle stability in real applications, it can be expected that such an EIR-like metamaterial will be highly useful to promote the development of slow light device.
Funding
platforms of PCSIRT Project (No. IRT_17R40); Science and Technology Program of Guangzhou (No. 2019050001); Guangdong Province Basic and Applied Research Fund (Grant No. 2019B1515120037).
Acknowledgments
We appreciate the financial support from the National 111 Project, and the MOE International Laboratory for Optical Information Technologies are also appreciated for providing technical support.
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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