Controlling the interaction between plasmon-induced transparency and guided mode resonance

Xiaolei Zhao; Cai Yuan; Yeyu Zhu; Xiangfeng Chen; Lin Zhu

doi:10.1364/OE.25.030043

1. Introduction

Electromagnetically induced transparency (EIT) effect, a result of the quantum destructive interference between different excitation pathways, was first observed in multi-level atomic systems [1,2]. This quantum effect can give rise to markedly enhanced nonlinear susceptibility and dramatic dispersion in the induced transparency region, making it very attractive for applications of nonlinear optics and slow light [3–5]. However, stringent experimental conditions and limited material options severely hinder the practical applications of quantum EIT systems as well as the implementation of on-chip EIT. Recently, along with the advances in metamaterial technology, the plasmonic analogue of EIT in metamaterial structure, also termed as plasmon induced transparency (PIT) effect, has attracted increasing attention [6–12]. Metamaterial-based PIT systems can exhibit EIT-like characteristics through flexible design and provide integrated plasmonic applications, while avoiding the stringent operating conditions of quantum EIT.

The PIT effect, generally, arises from the destructive interference between the bright (superradiant) and dark (subradiant) plasmon modes [6]. The bright mode strongly couples to the incident field and thus usually possesses a low quality (Q-) factor, whereas the dark mode only weakly or cannot directly couple to this external field, exhibiting a relatively high Q-factor. Owing to the inherent properties of metamaterials such as high radiation [13] and intrinsic heating losses [14], it is difficult to obtain high-Q transparency peaks in metamaterial-based PIT systems from terahertz to visible wavelengths. High-Q factors are often needed to obtain large group delay and enhanced slow light effect. Additionally, most PIT metamaterials reported up to now only have a single transparency peak. Recent discoveries of hybrid metamaterial-waveguide (HMW) systems [15–21] open promising ways to overcome these issues, showing significant applications ranging from filtering to optical information processing and multi-band slow light technology. It is well established that the regular HMW systems can achieve double transparency peaks with high Q-factors resulting from the destructive interferences between a broadband bright plasmon mode and two diffractively-excited, quasi-guided waveguide modes [19–21]. Nonetheless, it is still highly challenging to obtain three or more PIT peaks with high Q-factor within a single device.

In this letter, we propose a hybrid PIT metamaterial-waveguide (PITMW) system at mid-infrared (IR) wavelengths, which excites the quasi-guided modes through not only the diffracted waves but the near field coupling with the dark plasmon mode. The hybrid PITMW system shows the very intriguing transmission properties with multi-spectral transparency peaks due to the complex coupling mechanisms between the PIT effect and different quasi-guided modes. To our best knowledge, this paper presents the first in-depth results concerning the coupling between the quasi-guided waveguide mode and dark plasmon mode.

2. Results and Discussions

Figure 1 illustrates the schematic view of the proposed hybrid PITMW system, where a PIT metamaterial layer is placed on top of a dielectric slab waveguide. A plane wave propagating in the z direction is normally incident upon the hybrid system with the electric field polarized along the y direction. In the dielectric waveguide, the waveguide layer with the refractive index n_w is sandwiched between the cladding and the substrate, both of which have the refractive index n_c. n_c is assumed to be 1.41 throughout this paper, while n_w is at first assumed to be 1.90 for Fig. 2. The thicknesses of the waveguide and the cladding layers are assumed to be t_w = 2.8 µm and t_c = 3.0 µm, respectively. The PIT metamaterial layer is made of gold. Its unit cell consists of a vertical (y-oriented) cut wire (the dipole antenna), acting as the bright resonator, and a horizontal (x-oriented) cut wire pair (the quadrupole antenna), acting as the dark resonator. The simulated calculations in this letter are performed by means of the finite-difference time-domain (FDTD) method. The transmission spectrum of the only PIT metamaterial is shown as the red dashed curve in Fig. 2(a) with the geometrical parameters of the unit cell given in the caption. As expected, it shows a prominent PIT window, resulting from the destructive interference between the plasmonic dipole (bright) and quadrupole (dark) modes [22,23]. It it worth mentioning that all the geometrical and material parameters used in this paper can be further optimized to obtain the better performance or be modified to other wavelength bands.

Fig. 1 (a) Schematic of the hybrid PITMW system and the normally incident excitation. The top view (b) and the cross-section view (c) of the unit cell.

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Fig. 2 (a) Transmission spectra of the hybrid PITMW system (black solid line) and the PIT metamaterial (red dashed line). The geometrical parameters of the PIT unit cell: P_x = 6.6 μm, P_y = 6 μm, L = 3.6 μm, W = 1.4 μm, l = 3.1 μm, w = 0.9 μm, s = 1.6 μm, d = 0.4 μm. (b)-(d) Electromagnetic field distributions near the waveguide layer of the PITMW system at Peak 1 to 3: (b) distributions of |E|, E_y, |H|, H_x on the yOz cross section at 9.626 µm; (c) distributions of |E|, E_x, |H|, H_y on the yOz cross section at 9.980 µm; (d) distributions of |E|, E_y, |H|, H_x on the xOz cross section at 10.936 µm.

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The black solid curve in Fig. 2(a) represents the transmission spectrum of the hybrid PITMW system. It can be clearly observed that additional three transparency peaks (denoted as Peak 1 to 3) with high Q-factors appear in spectrum after the waveguide layer is embedded. These three peaks all arise from the excitation of the quasi-guided waveguide modes. By analyzing the field distributions at 9.626 and 10.936 µm shown in Figs. 2(b) and (d), we can identify that Peak 1 and Peak 3 are respectively attributed to the TM₀ quasi-guided mode propagating along the y direction (identified as TM_0,_y mode) and the TE₀ quasi-guided mode propagating along the x direction (TE_0,_x mode). Under the y-polarized incident radiation, the periodic arrangement of the metamaterial array can provide the necessary momentum to couple its diffracted waves into these two quasi-guide modes. And because of their destructive interferences with the dipole plasmon mode, the quasi-guided modes manifest themselves as sharp peaks in the transmission spectrum. Different from Peak 1 and 3, the origin of Peak 2 cannot be explained in this conventional way since it corresponds to the TE₀ quasi-guided mode propagating along the y direction (TE_0,_y mode) with the electric field polarized along the x direction, as shown in Fig. 2(c). This quasi-guided mode is excited through the near field coupling of the plasmonic quadrupole mode in the horizontal cut wire pair. Even though the quadrupole mode is sub-radiation (“dark”) under the y-polarized incidence, the electric field redistribution caused by the dipole antenna enables the generation of the x-component of the electric field, activating the quadrupole mode. The x-oriented plasmon polariton of the quadrupole mode further causes the excitation of TE₀ quasi-guided mode propagating along the y direction, giving rise to Peak 2. Please note that although it is possible to excite the TE_0,_y quasi-guided mode via the conventional pathway by simply changing the incident polarization to along x direction, it is impossible to simultaneously achieve three quasi-guided modes under this polarization condition due to the absence of the dark plasmon mode. It is the dark mode that plays the critical role in the unconventional excitation mechanism.

To further verify the distinction between the conventional and unconventional excitation pathways of the quasi-guided modes as well as the nature of Peak 1 to 3, the role of the waveguide structure is theoretically explored. The waveguide layer sandwiched between the cladding and the substrate can be treated as a slab waveguide that supports TE and TM quasi-guided modes. Following a standard analysis, it is shown that the quasi-guided modes need to satisfy the energy dispersion for the self-consistency [24–26]:

\tan (κ t_{w}) = \frac{κ (γ_{c} + γ_{s})}{κ^{2} - γ_{c} γ_{s}}, (TE mode)

\tan (κ t_{w}) = \frac{n_{w}^{2} κ (n_{s}^{2} γ_{c} + n_{c}^{2} γ_{s})}{n_{c}^{2} n_{s}^{2} κ^{2} - n_{w}^{4} γ_{c} γ_{s}}, (TM mode)

where

κ = \sqrt{n_{w}^{2} k_{0}^{2} - β^{2}}

,

γ_{c} = \sqrt{β^{2} - n_{c}^{2} k_{0}^{2}}

,

γ_{s} = \sqrt{β^{2} - n_{s}^{2} k_{0}^{2}}

, and k₀ = 2π/λ.

β = n_{e f f} \cdot k_{0}

is the propagation constant in the x-y plane, and n_eff is the effective refractive index of the waveguide mode. From Eqs. (1) and (2), we can obtain the dispersion relations of the TE_m and TM_m mode when n_w = 1.90, n_c = 1.41, and t_w = 2.8 μm. The subscript m represents the mth order of the guided modes. However, these guided modes cannot be excited by an externally incident wave, because the momentum and energy cannot be conserved simultaneously. When the metamaterial layer is placed on the slab waveguide, the periodic arrangement of the unit cells can provide the necessary momentum to couple the diffracted waves into the guided modes. Under this circumstance, these guided modes are no longer fully confined within the waveguide layer, but leaky in the outer region, becoming quasi-guided modes. In our design, the necessary momentums to excite these quasi-guided modes are provided by the PIT metamaterial array with the periods of P_x = 6.6 μm and P_y = 6 µm. For efficient resonant coupling, the following phase matching condition needs to be satisfied [27,28]:

β = k_{x - y} + i \frac{2 π}{P_{x}} x + j \frac{2 π}{P_{y}} y

where k_x-y represent the component of incident wave vector in the x-y plane. i and j are the integers respectively labeling the orders of grating vectors in the x and y axis. Considering normal incidence (k_x-y = 0) and the first-order grating vector, Eq. (3) can be simplified as

β = 2 π / P_{x} x

or

β = 2 π / P_{y} y

. By inserting this condition to the dispersion relations, the theoretical resonant wavelengths of the quasi-guided modes can be obtained: λ_TM0,_y = 9.625 µm, λ_TE0,_y = 10.052 µm, λ_TE0,_x = 10.937 µm, which are marked as the orange crosses in Fig. 2(a). Note that, since the TM_0,_y and the TE_0,_x quasi-guided modes are directly excited from the free-space waves via the diffractive coupling, the above theoretical analyses are expected to well predict their resonance positions. One can clearly see that λ_TM0,_y and λ_TE0,_x are well matched with the resonant wavelengths of Peak 1 and 3. However, the excitation pathway of the TE_0,_y quasi-guided mode is rather more complex. The near-field coupling from the dark plasmon mode plays the key role here. It is through this coupling that the dark mode and TE_0,_y can interact with each other, which induces the splitting of resonance modes and eventually leads to the resonant wavelength shifts. Consequently, Peak 2 slightly blue shifts relative to its theoretical value λ_TE0,_y, and Peak 0 shows a small red shift from the previous PIT window, as illustrated in Fig. 2(a).

Figures 3(a)–3(c) illustrate the transmission spectra of the hybrid PITMW system for various values of n_w. Now we focus on the transmission dip (indicated as the grey area and identified as TD_TE_y), which is caused by the interaction between the dark plasmon mode and the TE_0,_y quasi-guided mode. The pink dots in the spectra indicate the points at which the strongest electric field intensity around the waveguide layer locates, and the corresponding distribution patterns are shown as the insets of Figs. 3(a)–3(c). These distribution patterns clearly demonstrate the excitation of the TE_0,_y quasi-guided mode. Besides, it can be observed that the pink dot gradually shifts to the longer wavelengths as n_w increases, which indicates the red-shift of the TE_0,_y mode. As the spectral detuning between the TE_0,_y mode and the PIT window gradually decreases to zero, TD_TE_y gradually transforms from an asymmetric Fano line-shape into a symmetric Lorentzian one. The electric field distributions in the PIT metamaterial layer are also investigated to better understand the interaction between the dark plasmon mode and the TE_0,_y quasi-guided mode. Figure 3(d) represents the |E| distribution on the xOy plane of the PIT metamaterial at 10.500 µm indicated by the blue vertical line. Without the waveguide layer, one can observe that strong electric fields concentrate around the dark resonator. However, when the waveguide layer (n_w = 2.00) is embedded, as shown in Fig. 3(e), the destructive interference between the dark plasmon mode and the TE_0,_y quasi-guided mode obviously suppresses the excitation of the dark plasmon mode, which ultimately weakens the electric fields around the dark resonator but strengths the fields around the bright resonator. Accordingly, the spectrum exhibit a prominent transmission dip at 10.500 µm owing to the relatively large radiation loss brought by the bright mode.

Fig. 3 Interaction between the dark plasmon mode and the TE_0,_y mode. (a-c) Transmission spectra of the hybrid PITMW system (solid lines) for various values of n_w from 1.92 to 2.00. The other geometrical parameters are same as before. The insets represent |E| distributions on the yOz plane at the wavelengths indicated by the pink dots. (d) and (e) respectively represent |E| distributions on the xOy plane of the PIT metamaterial and the PITMW system with n_w = 2.00 at 10.515 µm indicated by the blue vertical line.

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Next, we investigate the interaction between the PIT window and the TE_0,_x quasi-guided mode. To overlap them in spectrum, the geometrical parameters of the PIT metamaterial are slightly adjusted with the specific values provided in the caption of Fig. 4. Transmission spectra of the hybrid system with n_w from 1.82 to 1.94 are studied and plotted in Figs. 4(a)–4(e). As n_w increases, the TE_0,_x quasi-guided mode also undergoes a red-shift, which can be demonstrated by the pink dots in the transmission spectra and the corresponding |E| distributions shown in the insets. Additionally, we can clearly observe that, as the TE_0,_x modegets closer to the central wavelength of the PIT window, the transmissivity and the Q-factor of the transmission dip TD_TE_x (indicated as the grey area in Figs. 4(a)–4(e)) greatly increases. Besides, TD_TE_x always keeps the asymmetric Fano lineshape during this change process. By comparing the evolution of TD_TE_x with that of TD_TE_y, it can be concluded that the TE_0,_x mode has a much weaker coupling with the PIT window than the TE_0,_y mode. This further confirms the origin of the TE_0,_y mode that is attributed to the dark plasmon mode with the same electric field polarization, while the TE_0,_x mode is not. Therefore, the TE_0,_y mode can directly couple with the dark plasmon mode through the near field interaction; however, the TE_0,_x mode is only indirectly coupled with the dark plasmon mode via the bright plasmon mode. In another perspective, this indirect coupling between the TE_0,_x mode and the dark plasmon mode can also be interpreted as the interaction between two “dark” modes. Because the coupling between them are indirect and thus relatively much weaker, the transmission dip TD_TEx shows a much narrower linewidth than TD_TEy. Note that the coupling between the PIT window and the TE_0,_x mode leads to a rapid change of transmissivity within TD_TE_y. For example, when n_w = 1.85, the transmissivity decreases from 0.953 to 0.131 with a small increase in wavelength from 10.701 to 10.723 µm. To better understand this effect, we further analyze the electric field distributions near the metamaterial layer. Figures 4(f) and 4(g) respectively show the |E| distributions of the PIT metamaterial and the PITMW system (n_w = 1.85) at 10.699 µm indicated by the green triangle. We can clearly see that electric fields around the bright and dark resonators are both significantly suppressed after the waveguide layer is added. This is consistent with the high transmissivity at this wavelength. However, at the dip (10.723 µm) indicated by the blue triangle, as illustrated by Fig. 4(i), electric fields are greatly enhanced around both the bright and dark resonators. The high loss from the bright plasmon mode leads to the reduced transmissivity.

Fig. 4 Interaction between the PIT window and the TE_0,_x mode. (a-e) Transmission spectra of the hybrid PITMW system (solid lines) for various values of n_w. The geometrical parameters of the PIT unit cell are P_x = 6.6 μm, P_y = 6 μm, L = 3.8 μm, W = 1.4 μm, l = 3.2 μm, w = 0.9 μm, s = 2 μm, d = 0.4 μm. The insets represent |E| distributions on the xOz plane at the wavelengths indicated by the pink dots. (f-g) represent |E| distributions on the xOy plane of the PIT metamaterial and the PITMW system (n_w = 1.85) at 10.699 µm indicated by the green triangle. (h-i) represent |E| distributions on the xOy plane of the PIT metamaterial and the PITMW system (n_w = 1.85) at 10.723 µm indicated by the blue triangle.

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3. Conclusion

In conclusion, we have investigated a hybrid PITMW system where a PIT metamaterial array is utilized to excite the quasi-guided modes in the underneath waveguide layer. The hybrid system supports three quasi-guided modes (TM_0,_y, TE_0,_y and TE_0,_x) under the y-polarized normal incidence. It is shown that TM_0,_y and TE_0,_x are directly excited from the diffracted waves of free-space incidence, while TE_0,_y is induced by the x-oriented plasmon polariton of the dark plasmon mode. Owing to the interactions between the three quasi-guided mode and the PIT effect, the PITMW spectrum exhibits intriguing transmission characteristics with multispectral induced transparency windows. The strong coupling between TE_0,_y and the dark mode results in a prominent transmission dip within the PIT window. On the other hand, TE_0,_x, or TM_0,_y, can only interact with the dark mode via the bright plasmon mode, which results in relatively weak coupling. Accordingly, the transmission dip caused by this interaction shows higher Q-factor and always keep the asymmetric Fano lineshape. Our findings offer in-depth understanding of the underlying physical mechanism in the hybrid PITMW system. Additionally, on account of the multispectral high-Q induced-transparency windows resulting from the complex coupling mechanisms, the proposed hybrid system is very promising for the development of multi-band slow light devices and sensitive biosensor at mid-IR.

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Controlling the interaction between plasmon-induced transparency and guided mode resonance

Abstract

1. Introduction

2. Results and Discussions

3. Conclusion

References and links

Cited By

Figures (4)

Equations (3)

Optics Express