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Abstract
Slow light technique has significant potential applications in many contemporary photonic device developments for integrated all-optical circuit, such as buffers, regenerators, switches and interferometers. In this paper, we present an efficient coupling mechanism of an electromagnetically induced transparency like (EIT-like) effect in an all-dielectric nano-metamaterial. This EIT-like effect is generated by destructive interference between a radiative Fabry-Perot (FP) mode and a dark waveguide (WG) mode, which is based on a combined structure of a dielectric grating and multilayer films. The dark WG mode is excited by guided mode of dielectric grating instead of radiative FP mode. In analogy to the molecular transition process, the FP mode, guided mode and WG mode are denoted by excited states of |1〉, |2〉 and |3〉. The two coupling pathways of the EIT-like effect in our metamaterial are |0〉 → |1〉 and |0〉 → |2〉 → |3〉 → |1〉, where |0〉 is the ground state. The simulated resonant wavelength of WG mode is consistent with theoretical result. We further confirm this EIT-like effect through a two-oscillator coupling analysis. We achieve a group refractive index of 913.6 by adjusting these two modes coupling of the EIT-like effect, which is useful for developing slow light device. This work provides a valuable solution to realize electromagnetically induced transparency in all-dielectric nanomaterial.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The velocity of light in vacuum is approximately $3 \times {10^8}\;\textrm{m/s}$. This ultrahigh speed provides tremendous advantage for optical data transmission, but also limits the optical signal controlling in time domain. To overcome this bottle-neck issue, many possible technical solutions have been investigated to scale down the group velocity of a compound light pulse consisting of different frequencies, that is, to obtain slow light. Currently, many slow light techniques are investigated for developing high efficient photonic devices, such as buffers [1–3], regenerators [4,5], switches [6,7] and interferometers [8,9]. There are some ordinary solutions to generate slow lights, which include electromagnetically induced transparency (EIT) [10–12], coherent population oscillation [13,14], optical parametric amplification [15,16], stimulated Brillouin scattering [17,18] or photonic crystal based method [19,20]. All these methods aim at creating a narrow spectral region with changes, exactly, peaks in the dispersion relation. EIT is one of the most promising way to achieve low group velocity of light.
An EIT phenomenon was initially observed in a three-level atomic system [21] with ground state of $|0 \rangle$ and excited states of $|1 \rangle$ and $|2 \rangle$. A probe light excites the molecular transition from $|0 \rangle$ to $|1 \rangle$. When frequency of a second driven light satisfies the molecular transition from $|1 \rangle$ to $|2 \rangle$, destructive interference occurs between the two transition processes and the population of state $|1 \rangle$ is nearly zero, which generate a transparency window in spectrum. Using EIT, L. V. Hau et al. demonstrated a group velocity of 17 m/s in an ultra-cold gas of sodium atoms experimentally, which is acknowledged as the first realization of slow light [10]. D. F. Phillips et al. reported the “storage of light” by dynamically reducing the group velocity of the light pulse to zero in a vapor of rubidium atoms [22]. B. Wu et al. observed optical pulse delays of 16 ns with a delay-bandwidth product of 0.8 in a planar chip consisting of hot rubidium atoms in hollow-core waveguides [23]. Nevertheless, atomic EIT systems suffer from extreme experimental conditions, such as temperature, which make it with high cost and not convenient for device integration. Therefore, EIT-like effect at room temperature in solid systems, analogy of EIT in atomic system, has attracted great attentions. K. Totsuka et al. performed an EIT-like effect in a fiber taper by coupling two ultrahigh-Q silica microspheres with different diameters, and observed a 8.5 ns delay for a 51 ns pulse width [24]. J. Gu et al. presented an optically tunable group delay of ultrafast optical pulses at terahertz by integrating photoconductive silicon into the metamaterial unit cell and generating EIT-like effect [25]. The EIT-like effect was also reported in nano plasmonics nanasystem, which is named as plasmon induce transparency (PIT), referring to a EIT-like effect induced by surface plasmon polaritons (SPPs). S. Zhang et al. investigated a PIT formed by a dipole antenna and two strips in a plasmonic metamaterial [26]. R. Taubert et al. demonstrated a PIT with the coupling of a broad dipolar and a narrow dark quadrupolar plasmons [27]. T. T. Kim et al. achieved an electrically tunable graphene EIT metamaterial at THz regime [28]. J. Hu et al. investigated the coupling of graphene Tamm plasmon polaritons (TPPs) and silver TPPs in a graphene/dielectric Bragg reflector/Ag slab hybrid system [29]. Whether in these dielectric or plasmonic nanosystems, the physical mechanisms of EIT-like effect are slightly different from that in the three-level atomic system. The EIT-like effect is formed by destructive interference of a radiative mode and a dark mode. The dark mode is excited by radiative mode instead of external field directly. If we analogize the EIT-like effect in atomic system, the radiative and dark modes are designated as two excited states $|1 \rangle$ and $|2 \rangle$ respectively. The EIT-like effect is formed the coupling of two pathways of $|0 \rangle \to |1 \rangle$ and $|0 \rangle \to |1 \rangle \to |2 \rangle \to |1 \rangle$. Recently, the EIT-like effect is also generated in nanoscaled all-dielectric nanosystem. S.-G. Lee et al. presented an EIT-like effect by coupling two resonant guide modes with a low- and high-quality factor [30]. C. Sui et al. generated the EIT-like effect by using destructive interference between a broad magnetic resonance of a dipole and a narrow guide mode resonance in the substrate [31]. B. Han et al. presented a EIT-like effect by coupling toroidal dipole moment and magnetic resonance with an E-shaped silicon array [32]. Although the EIT-like effects in all-dielectric nanosystems have been reported, efforts are still required to further reveal underlying mechanism of EIT-like effect for practical applications.
In this paper, we present an efficient coupling mechanism of an EIT-like effect in an all-dielectric nano-metamaterial based on combined grating and multilayer structure. We study the EIT-like effect by using finite difference time domain (FDTD) method and two-oscillator coupling theory. The EIT-like effect is formed by destructive interference of a radiative Fabry-Perot (FP) mode and a dark waveguide (WG) mode. The dark WG mode is excited by guided mode of dielectric grating instead of radiative FP mode. If the FP mode, guided mode and WG mode are represented by the excited states of $|1 \rangle$, $|2 \rangle$ and $|3 \rangle$, the two coupling pathways of the EIT-like effect in our metamaterial are $|0 \rangle \to |1 \rangle$ and $|0 \rangle \to |2 \rangle \to |3 \rangle \to |1 \rangle$. The simulated resonant wavelength of WG mode is consistent with theoretical result, which support our explanation of the origin of the EIT-like effect. We also achieve a maximum group refractive index of 913.6 by using this metamaterial mechanism, which can be potentially applied for slow light device development.
2. Model and theory
Figure 1 shows the proposed all-dielectric nano-metamaterial. It includes a grating with ZnO-dopted SiO2 material, a Cytop fluoropolymer layer, a ZnO-dopted SiO2 layer, and a Cytop fluoropolymer substrate. Figure 1(a) is the whole device schematics, and Fig. 1(b) is the profile view. The dielectric grating is defined by its period P, slit width a, and thickness t. The thickness of Cytop fluoropolymer layer and ZnO-dopted SiO2 layer are described by ${d_1}$ and ${d_2}$, respectively. A p-polarized light is illuminated from grating side with a wavevector ${k_0}$ and an angle of $\theta $ to the y axis.
Fig. 1. Schematic illustration of the proposed all-dielectric nano-metamaterial: (a) whole schematics and (b) profile view.
A two-dimensional FDTD method is adopted for the study in this paper. The mesh size is ${5}\;\textrm{nm} \times {5}\;\textrm{nm}$, which is small enough to deal with the problem of electromagnetic wave propagation in dielectric material. The periodic boundaries and perfectly matched layers (PML) are applied on x and y boundaries of the simulation space. The permittivity of Cytop fluoropolymer material is set as experimental data with $\varepsilon = n_1^2 = 1.8117 + i2.6900 \times {10^{ - 3}}$ [33]. The refractive index of ZnO-dopted SiO2 material is ${n_2} = 2.198$.
There are three types of resonances in this all-dielectric nano-metamaterial: FP mode in the slit of the dielectric grating, the guided mode of the grating and the WG mode in the ZnO-dopted SiO2 layer. In the dielectric grating slit, the resonant condition of the FP mode satisfies
where ${k_{\textrm{FP}}}$ is the wavenumber of the FP mode, m is an integer representing the order of the FP mode, and ${\rho _1}$ and ${\rho _2}$ are the reflection coefficients at the two ends of the slit.The propagating constant of the guided mode of dielectric grating contains two parts: parallel wavevector component of incident light and wavevector component provided by the grating. It is expressed as
where $G = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } P}} \right.} P}$, and n is an positive integer of $1{,_{}}2{,_{}}3 \cdots$.A symmetric three-layered waveguide is formed by the Cytop fluoropolymer layer, the ZnO-dopted SiO2 layer, and the Cytop fluoropolymer substrate. In the waveguide, since we perform a two-dimensional simulation with electric field paralleled to the simulation plane, only TM mode of the WG mode can be excited. The dispersion relation of TM mode is expressed as [34]
In this designed nano-metamaterial structure, the FP mode and guided mode are excited directly as light interacts with the grating. When the propagating constant of guided mode matches that of the WG mode, the WG mode is excited subsequently. As light propagating continuously, the transmission and reflection spectra contain the information of coupling of FP mode and WG mode.
We will explain the coupling mechanism of EIT-like effect in section 3.1. The EIT-like effect is generated by destructive interference between a radiative FP mode and a dark WG mode, where the FP mode describes the broad peak and the dark WG mode determines the narrow dip of EIT-like effect. By solving Eqs. of (2) and (3), we can theoretically obtain the dip wavelength of EIT-like effect (i.e. resonant wavelength of the WG mode).
3. Results and discussions
We will discuss the coupling mechanism of EIT-like effect in detail in Section 3.1. Then, the EIT-like effect is further analyzed with a two-oscillator coupling theory for slow light application in Section 3.2.
3.1 Physical mechanism of the EIT-like effect
We start our study from analyzing reflection and transmission responses of the all-dielectric nano-metamaterial with different grating thickness t under a TM polarized light, as shown in Figs. 2(a) and 2(c). The geometrical parameters of the metamaterial are defined as $P = {500}\;\textrm{nm}$, $a = {100}\;\textrm{nm}$, ${d_1} = {700}\;\textrm{nm}$ and ${d_2} = {100}\;\textrm{nm}$. A normally incident plane wave is considered. It is found that a narrow mode interferes with a broad mode for TM polarization. Then, we extract the reflection spectra with grating thickness of $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$ as illustrated in Figs. 2(e) and 2(f), which are indicated in black dashed lines in Fig. 2(a). A EIT-like effect is exhibited for $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$. The full width at half maximum (FWHM) of the EIT-like effect dip is 2 nm for $t = {160}\;\textrm{nm}$ and 1.7 nm for $t = {525}\;\textrm{nm}$. Its peak and dip wavelengths of the EIT-like effect are also illustrated in red in the two subfigures.
Fig. 2. Reflection spectra of all-dielectric nano metameterial with different t under (a) TM and (b) TE polarized light; Transmission spectra with different t under (c) TM and (d) TE polarized light; Reflection spectra with (e) $t = {160}\;\textrm{nm}$ and (f) $t = {525}\;\textrm{nm}$ under TM polarization; (g) Transition levels of the EIT-like effect in all-dielectric nano-metamaterial.
To reveal the physics mechanism of the EIT-like effect, the detailed near-field distributions are given out. Figures 3(a)–3(c) illustrate electromagnetic field distributions of the peaks and dip of the EIT with $t = {160}\;\textrm{nm}$, while Figs. 3(d)–3(f) display the corresponding results with $t = {525}\;\textrm{nm}$. Taking $t = {160}\;\textrm{nm}$ as an example, a 1st order FP mode is formed in the grating slit at $\lambda = 721.8\;\textrm{nm}$ and $\lambda = 728.8\;\textrm{nm}$, as shown in Figs. 3(a) and 3(c). These two wavelengths correspond to the peaks of EIT-like effect. Figure 3(b) shows that a WG mode is excited in the ZnO-dopted SiO2 layer at the EIT dip of $\lambda = 725.0\;\textrm{nm}$. For $t = {525}\;\textrm{nm}$, Figs. 3(d)–3(f) illustrate the similar results except that the FP mode is 2nd order. These detailed distributions explain that the EIT-like effect is formed by the coupling of the 1st or 2nd order FP mode and WG mode. The theoretical resonant wavelength of the WG mode is 724.9 nm, which is derived from Eqs. (2) and (3). The simulated result is 725.0 nm, which is consistent with the theory. This consistency demonstrates that the WG mode is excited by guided mode. Figures 2(b) and 2(d) illustrate reflection and transmission responses with different t under a TE polarization. There is no structure paralleled to the electric field direction for TE polarization, therefore, no TE mode of WG mode is excited. As a result, the EIT-like effect is not shown for TE polarization.
Fig. 3. Electric and magnetic field distributions of (a) $t = {160}\;\textrm{nm}$, $\lambda = {721.8}\;\textrm{nm}$; (b) $t = {160}\;\textrm{nm}$, $\lambda = {725.0}\;\textrm{nm}$; (c) $t = {160}\;\textrm{nm}$, $\lambda = {728.8}\;\textrm{nm}$; (d) $t = {525}\;\textrm{nm}$, $\lambda = {722.4}\;\textrm{nm}$; (e) $t = {525}\;\textrm{nm}$, $\lambda = {725.0}\;\textrm{nm}$; and (f) $t = {525}\;\textrm{nm}$, $\lambda = {728.2}\;\textrm{nm}$.
In the metamaterial, the broad FP mode is supported by the dielectric material with a positive real part in permittivity at optical frequency. Therefore, the FP mode acts as a broad peak in the reflection, as shown in Fig. 2(a). When a broad FP peak couples with a narrow WG mode, the narrow EIT dip is formed, as shown in Figs. 2(e) and 2(f).
To explicitly reveal the origin of the EIT-like effect, we analogize the EIT to molecular transition process, as shown in Fig. 2(g). The ground state is displayed by $|0 \rangle$. The FP mode, guided mode and WG mode are represented by the excited states of $|1 \rangle$, $|2 \rangle$ and $|3 \rangle$. The radiative FP mode is directly excited in the slit of the dielectric grating when the external field of light matches the FP mode condition. It corresponds to the transition pathway of $|0 \rangle \to |1 \rangle$. The WG mode cannot be excited by incident light, which means that the pathway of $|0 \rangle \to |3 \rangle$ is forbidden. The direct coupling between the FP mode and WG mode is also impossible because their propagations are in perpendicular directions. Thus, the pathway of $|0 \rangle \to |1 \rangle \to |2 \rangle \to |1 \rangle$ cannot be realized. The dark WG mode is excited by the guided mode of the dielectric grating, which has already been demonstrated by the consistency between the simulated and theoretical wavelengths of WG mode. Thus, the second transition pathway is $|0 \rangle \to |2 \rangle \to |3 \rangle \to |1 \rangle$. An EIT-like effect is formed by the coupling of the two transition pathways.
Subsequently, we focus on the influence of the grating period P. Figures 4(a) and 4(b) show reflection spectra of the metamaterial with different grating thicknesses t for $P = {600_{}}{\mathop{\textrm {nm}}\nolimits} $ and $P = {700}\;\textrm{nm}$. A coupling between FP mode and WG mode is exhibited for these two periods. The resonant wavelength of WG mode redshifts because larger P will result in a smaller $\beta$, then, smaller ${k_0}$ and lager $\lambda$ according to Eqs. (2) and (3). The simulated wavelengths of the WG mode are 851.8 nm and 979.6 nm, which match the theoretical results of 851.3 nm and 979.7 nm. Figures 4(c) and 4(d) illustrate the reflection spectra with a continuous change of P from 500 nm to 800 nm for $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$. As P increases, the WG mode redshifts out of the 1st order (or 2nd order) FP mode for $t = {160}\;\textrm{nm}$ (or $t = {525}\;\textrm{nm}$).
Fig. 4. Reflection spectra with different t for (a) $P = {600}\;\textrm{nm}$ and (b) $P = {700}\;\textrm{nm}$; Reflection spectra with different P for (c) $t = {160}\;\textrm{nm}$ and (d) $t = {525}\;\textrm{nm}$.
Then, we investigate the effect of the grating slit width a on spectral response. Figures 5(a) and 5(b) show the reflection spectra with different grating thickness t for $a = {200}\;\textrm{nm}$ and $a = {400}\;\textrm{nm}$. A coupling between FP mode and WG mode is observed for $a = {200}\;\textrm{nm}$. The 2nd order and 3rd order FP modes are more distinct for $a = {200}\;\textrm{nm}$ compared with the result for $a = {100}\;\textrm{nm}$ in Fig. 2(a). For $a = {400}\;\textrm{nm}$, the WG mode cannot be excited because the duty cycle of the ZnO-dopted SiO2 material in one grating period is 1:5, which is too small to support guided mode. The FP mode is also faded out for such a small duty cycle. Figures 5(c) and 5(d) exhibit the continuous change of a from 50 nm to 450 nm for $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$. For $t = {160}\;\textrm{nm}$, only 1st order FP mode couples with WG mode. For $t = {525}\;\textrm{nm}$, both 1st order and 2nd order FP modes couple with WG mode. The upper and lower red regions represent the 1st order and 2nd order FP modes, as displayed in Fig. 5(d). If the WG mode is excited, whether how a is changed, the simulated wavelengths of WG mode remain at 725.0 nm . The results agree with the theoretical equations of (2) and (3), that means the slit width has no effect on WG mode.
Fig. 5. Reflection spectra with different t for (a) $a = {200}\;\textrm{nm}$ and (b) $a = {400}\;\textrm{nm}$; Reflection spectra with a continuous changing a for (c) $t = {160}\;\textrm{nm}$ and (d) $t = {525}\;\textrm{nm}$.
To demonstrate the 1st order FP mode in the upper red region in Fig. 5(d), reflection spectra of $a = {300}\;\textrm{nm}$ is plotted in Fig. 6(a), with marked wavelengths of peaks and dip of EIT-like effect. Figures 6(b)–6(d) illustrate magnetic field distributions at the peaks and dip of EIT. It is found that EIT-like effect is generated by the coupling of 1st order FP mode and WG mode for $a = {300}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$.
Fig. 6. Reflection spectrum with $a = {300}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$ in Fig. 5(d); Magnetic field distributions of (b) $\lambda = {722.0}\;\textrm{nm}$, (c) $\lambda = {725.0}\;\textrm{nm}$, and (d) $\lambda = {727.2}\;\textrm{nm}$.
Afterwards, we also study the effect of the thicknesses of the Cytop fluoropolymer layer of ${d_1}$ and the ZnO-dopted SiO2 layer of ${d_2}$. Figures 7(a) and 7(b) illustrate reflection spectra with different ${d_1}$ and ${d_2}$ with $t = {160}\;\textrm{nm}$, while Figs. 7(c) and 7(d) display the corresponding results with $t = {525}\;\textrm{nm}$. It is found that ${d_1}$ determines the coupling extent of FP mode and WG mode. As ${d_1}$ increases, this coupling of the two modes becomes weaker and EIT dip becomes narrower. The EIT-like effect almost disappears for ${d_1}\;>\;{1000}\;\textrm{nm}$, which indicates the interaction distance of FP mode and WG mode generating EIT-like effect is smaller than 1000 nm. As ${d_2}$ increases, high orders of TM mode appear and they interfere with the 1st order FP mode for $t = {160}\;\textrm{nm}$ and 2nd order FP mode for $t = {525}\;\textrm{nm}$. If the resonant wavelength of TM mode is fixed at 725.0 nm, the theoretical ${d_2}$ of $\textrm{T}{\textrm{M}_0}$, $\textrm{T}{\textrm{M}_1}$ and $\textrm{T}{\textrm{M}_2}$ are 100 nm, 319.4 nm and 538.8 nm. The theoretical ${d_2}$ of $\textrm{T}{\textrm{M}_0}$, $\textrm{T}{\textrm{M}_1}$ and $\textrm{T}{\textrm{M}_2}$ are derived from Eq. (3) with $q\textrm{ = }0{,_{}}1{,_{}}2$. With these ${d_2}$ conditions, the EIT-like effect is also generated. The electric and magnetic field distributions of the $\textrm{T}{\textrm{M}_0}$ with ${d_2} = 100{}_{}\textrm{nm}$ for $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$ have already been shown in Figs. 3(b) and 3(e). Figures 8(a)–8(d) illustrate the electric and magnetic field distributions of $\textrm{T}{\textrm{M}_1}$ and $\textrm{T}{\textrm{M}_2}$ with ${d_2} = 319.4{}_{}\textrm{nm}$ and ${d_2} = 538.8{}_{}\textrm{nm}$ for $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$. The thicknesses t have also been indicated by white outlines in the subfigures. The magnetic field distributions clearly show that there are two maximums for $\textrm{T}{\textrm{M}_1}$ mode and three maximums for $\textrm{T}{\textrm{M}_2}$ mode along y direction in the ZnO-dopted SiO2 WG.
Fig. 7. Reflection spectra with different (a) ${d_1}$ and (b) ${d_2}$ for $t = {160}\;\textrm{nm}$; Reflection with different (c) ${d_1}$ and (d) ${d_2}$ for $t = {525}\;\textrm{nm}$.
Fig. 8. Electric and magnetic field distributions of $\textrm{T}{\textrm{M}_1}$ and $\textrm{T}{\textrm{M}_2}$: (a) $t = {160}\;\textrm{nm}$, ${d_2} = {319.4}\;\textrm{nm}$; (b) $t = {160}\;\textrm{nm}$, ${d_2} = {538.8}\;\textrm{nm}$; (c) $t = {525}\;\textrm{nm}$, ${d_2} = {319.4}\;\textrm{nm}$ ; (d) $t = {525}\;\textrm{nm}$, ${d_2} = {538.8}\;\textrm{nm}$.
It should be noted that a bright red region appears at $\lambda = {938.4}\;\textrm{nm}$ for $t = {525}\;\textrm{nm}$ with changing ${d_1}$ and ${d_2}$, which is different form the results for $t = {160}\;\textrm{nm}$. Therefore, to illustrate the mechanism, we choose four typical points of A, B, C and D, as denoted by white characters in Fig. 7. Figure 9 gives out the electric and magnetic field distributions of point A, point B, point C and point D. The parameters of these four points are marked in detail in the captions of Fig. 9. From Figs. 9(b) and 9(d), we can see that the resonance at $\lambda = {938.4}\;\textrm{nm}$ in Figs. 7(c) and 7(d) is due to the presence of the 1st order FP mode in the grating slit.
Fig. 9. Electric and magnetic field distributions at $\lambda = {938.4}\;\textrm{nm}$ with (a) point A: $t = {160}\;\textrm{nm}$, ${d_1} = {675}\;\textrm{nm}$, ${d_2} = {100}\;\textrm{nm}$; (b) point B: $t = {525}\;\textrm{nm}$, ${d_1} = {675}\;\textrm{nm}$, ${d_2} = {100}\;\textrm{nm}$; (c) point C: $t = {160}\;\textrm{nm}$, ${d_1} = {700}\;\textrm{nm}$, ${d_2} = {275}\;\textrm{nm}$; (d) point D: $t = {525}\;\textrm{nm}$, ${d_1} = {700}\;\textrm{nm}$, ${d_2} = {275}\;\textrm{nm}$.
We further examine the real part of refractive index (i.e. ${n_1}^\prime$) in the fluoropolymer layer and the fluoropolymer substrate. The reflection spectra are shown in Figs. 10(a) and 10(b). The EIT turns into a Fano lineshape as ${n_1}^\prime$ changes from 1.346 to 1.40 in the fluoropolymer layer and substrate. The Fano lineshape is asymmetric because the narrow WG mode couples with one edge of the FP mode. The WG mode redshifts as ${n_1}^\prime$ increases. The theoretical wavelengths of WG mode are 724.9 nm, 725.8 nm, 727.9 nm, 730.2 nm, 732.5 nm, 734.9 nm and 737.3nm, as marked by color lines at the bottom of Fig. 10. The simulated wavelengths of WG mode are 725.0 nm, 725.8 nm, 728.0 nm, 730.0 nm, 732.4nm, 734.6 nm, 737.0nm with the changing of ${n_1}^\prime$ in the fluoropolymer layer, while they are 725.0 nm, 725.8 nm, 728.0 nm, 730.2 nm, 732.4nm, 734.8 nm, 737.4nm if we change ${n_1}^\prime$ in the fluoropolymer substrate. The simulated results agree with theoretical results. A refractive index (RI) sensitivity is defined by $S = {{\Delta \lambda } \mathord{\left/ {\vphantom {{\Delta \lambda } {\Delta \textrm{n}}}} \right.} {\Delta \textrm{n}}}$. According to the definition, RI sensitivity of all-dielectric nano-metamaterial is 227.8 nm/RIU. Another important parameter for the metamaterial is Q-factor, which is expressed as $\textrm{Q - factor} = {\textrm{f} \mathord{\left/ {\vphantom {\textrm{f} {\Delta \textrm{f}}}} \right.} {\Delta \textrm{f}}}$. In this metamaterial design, we achieve a Q-factor of 453.1, which is much higher than the Q-factor in former reported plasmonic works [26,35,36]. The improvement of Q -factor is because that we overcome the large non-radiative ohmic loss in this all-dielectric metamaterial structure.
Fig. 10. Reflection spectra of all-dielectric nano-metameterial with different ${n_1}^\prime $ (a) in the fluoropolymer layer and (b) in the fluoropolymer substrate.
3.2 Two-oscillator coupling theory and slow light application
In the all-dielectric nano-metamaterial, EIT-like effect is formed by the destructive coupling of a radiative FP resonance and a dark WG mode. To analogize EIT-like effect, these two modes can be viewed as two “molecule” oscillators [26,27]. Namely, the coupling between the two modes is regarded as a two-oscillator coupling. To provide a quantitative description of our EIT-like effect, an analysis equation of Eq. (4) is introduced. The radiative oscillator of ${x_1}(t )$ oscillates with central frequency ${\omega _0}$, which strongly couples to an external field of ${E_0}{e^{ - i\omega t}}$. While the dark oscillator of ${x_2}(t )$ weakly couples to external field, therefore the driven field term on the right of the equal sign is zero. The central frequency of the dark oscillator is ${\omega _0} + \delta$, where $\delta$ illustrate the central frequency shift of the dark oscillator compared with the radiative oscillator. The $\kappa$ stands for the coupling coefficient between the two oscillators. And ${\gamma _1}$ and ${\gamma _2}$ represent the damping of the two oscillators respectively.
Fig. 11. (a) Reflection spectra, transmission spectra and (b) absorption spectra with ${d_1}$ changing from 450 nm to 650 nm with an interval of 50 nm, for $t = {160}\;\textrm{nm}$.
In FDTD simulation, the coupling between FP mode and WG mode is determined by ${d_1}$. Figures 7(a) and 7(c) show that this coupling becomes weaker as ${d_1}$ increases. In two-oscillator coupling theory, $\kappa $ stands for coupling coefficient between the two oscillators. Smaller $\kappa $ indicates a weaker coupling of two oscillators. As shown in Fig. 11(b), the value of $\kappa $ decreases as ${d_1}$ increases form 450 nm to 650nm, which schematically illustrates the consistency of FDTD simulation and two-oscillator coupling theory.
Few can deny the fact that the polarizability $\vec{P}$ of a molecule is proportional to complex susceptibility $\chi $ and external field $\vec{E}$. The complex susceptibility $\chi $ is expressed by $\chi = {\chi _\textrm{r}} + i{\chi _\textrm{m}}$. According to “molecular” hypothesis theory, the polarizability $\vec{P}$ of a radiative “molecule” oscillator is scale with its oscillation ${x_1}(t )$ [26]. Therefore, we can derive that real part of susceptibility is still proportional to the real amplitude of ${x_1}(t )$. Then, the ${\chi _\textrm{r}}$ is expressed as
Fig. 12. (a) ${\chi _\textrm{r}}$ and (b) ${n_\textrm{g}}$ with different ${d_1}$ for $t = {160}\;\textrm{nm}$.
In this paper, to quantitatively describe the EIT-like effect for slow light application, the group refractive index is extracted from ${n_\textrm{g}} = c \times ({{{\textrm{d}k} \mathord{\left/ {\vphantom {{\textrm{d}k} {\textrm{d}\omega }}} \right.} {\textrm{d}\omega }}} )$. According to these Refs. [30,37], the dispersion relation of the metamaterial is determined by the expression of
where $\phi - {\phi _0}$ represent simulated phase difference with and without the metamaterial. In our system, the effective propagation length is expressed by $L = 2 \times ({t + {d_1} + {d_2}} )$ in consideration of reflection. By differencing both sides of Eq. (9), we can obtain the following relation:Finally, we investigate spectra response of all-dielectic nano-metamaterial with multiple-grouped WG structures. Figure 13 shows reflection spectra with double-grouped and triple-grouped WG structures for $t = {160}\;\textrm{nm}$ and $t = {525}\;\textrm{nm}$. It is found that there are multiple EIT-like effect appeared when more grouped WG structures are introduced into the metamaterial. And the number of the dips are consistent with the number of the grouped WG structures. Specifically, there are two EIT dips for double-grouped WG structures and three dips for triple-grouped WG structures. The appearance of multiple EIT-like effect by simply changing the number of the grouped WG structures, which will be beneficial for further slow light application.
Fig. 13. Reflection spectra with double-grouped and triple-grouped WG structures for (a) $t = {160}\;\textrm{nm}$ and (b) $t = {525}\;\textrm{nm}$.
4. Summary
In summary, we present a novel coupling mechanism to realize EIT-like effect in an all-dielectric nano-metamaterial in this paper. The EIT-like effect is generated through destructive interference between a radiative FP mode and a dark WG mode. The dark WG mode is excited by using the guided mode of dielectric grating instead of radiative FP mode. Then, the EIT-like effect of the metamaterial is fully investigated by adjusting the grating thickness t, grating period P, slit width a, fluoropolymer layer thickness ${d_1}$, WG layer thickness ${d_2}$, and the refractive indexes of the fluoropolymer layer and substrate. The resonant wavelengths of WG mode with these changing parameters agree well with the theoretical results. The EIT-like effect is further studied with a two-oscillator coupling theory. To verify the application capability of this metamaterial design, a group refractive index of 913.6 is achieved theoretically, which can be further investigated for slow light application. This provides a useful method to realize EIT in all-dielectric nano-metamaterials.
Funding
National Natural Science Foundation of China (11405020, 61520106013, 61727816, 51661145025).
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