1. Introduction

Electromagnetically induced transparency (EIT) observed in atomic media results from a coherent interaction between the atomic levels and the applied optical fields [1]. EIT promises a variety of potential applications such as slow light propagation, transfer of quantum correlation, and nonlinear optical process [2–5]. The demanding experimental conditions required to observe the EIT effect hinder its practical application, which catalyzes an ongoing search for classical systems mimicking EIT. In recent years, theoretical analysis and experimental observations have revealed that the EIT-like phenomena can also occur in dielectric photonic resonator systems, which is known as coupled-resonator induced transparency [6–9].

It is noteworthy that controlling light on a small scale is very essential for highly integrated optics. Surface plasmon polaritons (SPPs) propagating along the metal-dielectric interface can be well confined by ultrasmall metal structures and break the diffraction limit [10,11]. Among the different plasmonic devices, metal-dielectric-metal (MDM) plasmonic waveguides are of particular interest, because they support modes with deep wavelength scale and an acceptable length for SPPs. Based on the unique feature of MDM waveguides, the plasmonic analogue of EIT observed in nanoscale plamonic resonator systems was theoretically predicted and experimentally demonstrated in recent researches [12–20]. Veronis et al. introduced systems composing of a periodic array of metal-dielectric-metal (MDM) stub resonators side coupled to MDM waveguide, and showed that there is a trade off between the slowdown factor and the propagation length of the supported optical mode [12,13]. The on-chip plasmonic analogue of EIT was realized by detuned Fabry-Perot (FP) resonators aperture-side-coupled to a MDM waveguide [15]. In [17], Lu et al. demonstrated an analog of EIT and multiple induced-transparency peaks in plasmonic systems consisting of multiple cascaded nanodisk resonators, aperture-side-coupled to MDM bus waveguides. However, very few comprehensive studies have been performed on plasmon-induced transparency in MDM waveguide with two stub resonators.

In this paper, we investigate the EIT-like spectral responses and slow-light effects in plasmonic system composed of two stub resonators side-coupled to MDM bus waveguides. The system is easy to be realized and different from the configuration reported in [15,17,21]. In particular, the formation and evolution mechanisms of the transparency response induced by direct coupling and indirect coupling have been accurately analyzed through the CMT and FP model, respectively.

2. Theoretical analysis and discussion of MDM waveguide with one stub resonator

Figure 1(a) shows the dispersion curves for TM mode [22] in MDM waveguide with width h = 100 nm. The blue curve is for the real part of the effective index of SPPs, and the green curve is for the imaginary part. The inset in Fig. 1(a) is MDM waveguide structure with width h. The medium of the slit is assumed to be air (n = 1), and the background material in yellow is silver. The permittivity of silver is characterized by Drude model ε(ω) = 1-ω_p²/(ω² + iωγ_p), with ω_p = 1.38 × 10¹⁶ Hz and γ_p = 2.73 × 10¹³ Hz. These parameters are obtained by fitting the experimental results [23].

 

Fig. 1 (a) Blue and green curves represent the real and imaginary parts of the effective index n_eff of SPPs mode in MDM waveguide with width h = 100 nm, respectively. (b) Transmission spectra for MDM waveguide coupled to one stub, with w equaling to 100 nm and L being 500 nm. In both figures, insets display the corresponding schematic figures.

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Figure 1(b) shows the transmission spectra for the MDM waveguide with one stub resonator. The inset in Fig. 1(b) displays the schematic of the system. The FDTD method and perfectly matched layer boundary conditions are used in the numerical experiment. From the CMT [24,25], the transmission of the system supporting a resonant mode of frequency ω₀ can be expressed as

3. Plasmon-induced-transparency with two stub resonators

3.1 Plasmon-induced transparency by direct coupling

Figure 2(a) is a schematic illustration of MDM waveguide coupled to two stub resonators. When d is small, the direct and indirect couplings coexist between the two stub resonators, but the indirect coupling does not have obvious influence on the resonant frequencies of the coupled system. So, we mainly consider the direct coupling, as shown in Fig. 2(b). Using the CMT [24,25], the energy amplitudes a₁ and a₂ for the resonators of resonant frequencies ω₁ and ω₂ can be described by

 

Fig. 2 (a) Schematic of MDM waveguide coupled to two stub resonators: w, the width of the waveguide; w₁ and w₂, the widths of the stub resonators; L₁ and L₂, the stub depths; d, the inter-space between the two stub resonators. (b) Illustration of direct coupling between two resonators denoted by R₁ and R₂. k₁₂ and k₂₁ are coupling coefficients.

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Based on Eqs. (1) and (4), we investigate the EIT-like transmission characteristics of MDM waveguide coupled to two stub resonators. Figure 3(a) represents the transmission spectra for the structure in Fig. 2(a) at different inter-spaces d, while the other parameters are set as w = w₁ = w₂ = 100 nm, and L₁ = L₂ = 500 nm. The transmission dips correspond to the resonant frequencies (ω₊ and ω_-) of the coupled system. When d is greater than 100 nm, the direct coupling between the two identical resonators is weak, and the resonant frequency ω₊ is approximately equal to ω_-, so the EIT-like responses do not appear. As d decreases from 100 nm, the transparent window presents and becomes increasingly evident, which is due to the increment of coupling coefficient between the two stub resonance modes and is in accordance with the CMT.

 

Fig. 3 (a) Transmission spectra of MDM waveguide coupled to two stub resonators at different inter-spaces d. (b) The FWHM and group index of the transparent window versus d. (c-e) Magnetic field distributions at three wavelengths marked by triangles corresponding to d = 100nm, 50 nm, and 25 nm, respectively. The other parameters are changeless, namely, w = w₁ = w₂ = 100 nm, L₁ = L₂ = 500 nm.

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The measured full width at half maximum (FWHM) of the transparent window shown in Fig. 3(a) extends from 4 nm to 100 nm as d changes from 95 nm to 25 nm (plotted by blue dots in Fig. 3(b)). The corresponding group index of the transparent window for different inter-spaces d (plotted by red dots in Fig. 3(b)) can be approximately evaluated from the relation n_g = λ²/(4tΔλ) [26], where t is the propagation length in waveguide and Δλ is the FWHM of the transparent window. Combining Fig. 3(a) and Fig. 3(b), we can see that there is a trade-off between transmission ratio and group index. The largest group index of ~61 associated with a group velocity of ν_g ≈0.0164c is achieved when d = 95 nm, which is promising for the development of ultra-compact optical buffers. Such a system comprised of MDM waveguide side-coupled to a periodic array of the two direct coupling stub resonators can also be used to obtain slow light [12].

To get more insight into the physics of the observed EIT-like transmission, Figs. 3(c)-3(e) show the magnetic field distributions corresponding to three characteristic wavelengths marked by triangles for d = 100 nm, 50 nm and 25 nm, respectively. The magnetic field distributions indicate that the destructive interference of electromagnetic fields from the two resonators results in the EIT-like optical responses, which is similar to what happens in metamaterial-induced transparency. As d = 100 nm in Fig. 3(c), the stub on the right seems to make no difference, but the quality factor at resonant wavelength λ = 540 nm is smaller than that in Fig. 1(b), which is attributed to the increase of intrinsic loss.

In Fig. 4(a), we plot the transmission spectra for the system shown in Fig. 2(a), as w = w₁ = 100 nm, d = 50 nm, L₁ = L₂ = 500 nm, and w₂ = 60, 80, 100, 120, 160 and 170 nm, respectively. Figure 4(b) shows the FWHM and group index of the transparent window in Fig. 4(a). The FWHM (group index) of the transparent window decreases (increases) when w₂ varies from 40 nm to 100 nm, and then increases (decreases) as w₂ expands in the range of 100 – 160 nm. Consequently, for a fixed coupling distance in the direct coupling case, decreasing the difference between the two stub resonators gives rise to more pronounced slow-light effect. Comparing Fig. 3(b) with Fig. 4(b), we also find that the FWHM and group index to the inter-space between the two stub resonators are more sensitive than to the width of one cut, which provides an effective guidance for realizing the EIT-like phenomena. Figure 4(c) shows resonant frequencies (ω₊ and ω_-) of the system versus ω₂ using the CMT (solid curves) and FDTD method (dots). ω₂ is the resonant frequency of individual stub resonator and the other is ω₁ = 3.575 PHz. We adjust ω₂ by w₂ and have k₁₂ = k₂₁ = 1.4 × 10¹⁴ Hz. With the increment of ω₂ from 2.8 PHz to 4 PHz, resonant frequencies (ω₊ and ω_-) of the system exhibit a blue-shift. The resonant frequency difference Δω increases first and then decreases, which gives rise to the variation of transparent window in Fig. 4(a). As mentioned above, the formation and evolution mechanisms of plasmon-induced transparency by direct coupling have been proposed and accurately analyzed by the CMT, which is a powerful tool for the design of EIT-like systems.

 

Fig. 4 (a) Transmission spectra of the structure shown in Fig. 2(a) for different w₂, and the other parameters are set as follows: w = w₁ = 100 nm, d = 50 nm, and L₁ = L₂ = 500 nm. (b) The FWHM and group index of the transparent window versus w₂. (c) Resonant frequencies (ω₊ and ω_-) of the system calculated using the CMT (solid curves) and FDTD method (dots).

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3.2 Plasmon-induced transparency by indirect coupling

In this part, we introduce the indirect coupling scheme shown in Fig. 2(a) by referring to [17,21]. The parameter d is tunable and large enough that the two stub resonators do not interact with each other directly. All the other parameters of the structure are set as follows: w = w₁ = w₂ = 100 nm, L₁ = 500 nm, L₂ = 480 nm. Figure 5(a) shows the transmission spectra for different inter-spaces d. The curves with circles and triangles are the transmission spectra of the system with individual stub resonator (shown in Fig. 1(b)) comprising the coupled system. As expected, the transparent band is between the two individual stub resonator resonances. Moreover, the transparent resonance peak exhibits a shift and its symmetry is tunable.

 

Fig. 5 (a) Transmission spectra of MDM waveguide coupled to two stub resonators at different inter-spaces d with w = w₁ = w₂ = 100 nm, L₁ = 500 nm, L₂ = 480 nm. The curve with circles is transmission spectra for the one stub system (shown in Fig. 1(b)) with w = 100 nm, L = 480 nm, and the curve with triangles corresponds to the one stub system with w = 100 nm, L = 500 nm. (b) Transmission spectra for different inter-spaces d at λ = 527 nm. (c) Evolution of the transmission spectra versus δ and λ. (d) The top view of Fig. 5(c).

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The spacing dependence of the transparent window indicates the indirect coupling system as a two-mirror FP resonator [17, 21], in which the waveguide region between the two stub resonators acts as a FP cavity. The transmittance of the indirect coupled system can be given as

From Fig. 1(a) and Eq. (6), we can obtain that Re(n_eff) (the real part of the effective index of SPPs) approximately equals 1.254 and the periodicity of transmittance is 210 nm for λ = 527 nm. Using the FDTD simulations, we plot the transmission spectra in Fig. 5(b), in which the transmittance at λ = 527 nm varies periodically and the periodicity is 200 nm approximately. The slight difference in the periodicity between the FP model and FDTD method is caused by the inaccuracy of the propagation constant or the finite mirror size in the structure. The maximum of transmission spectra at λ = 527 nm gradually becomes smaller, which is due to the increase of the intrinsic loss. It is also noteworthy that the peak (dip) transmission at λ = 527 nm corresponds to phase retardation condition βδ = Nπ (βδ = (N + 1/2)π), for which the magnetic field distributions (not shown here) are consistent with [21] and N is an integer.

In Fig. 5(a), the resonant wavelengths for the two individual stub resonators are λ = 538.4 nm and 518.7 nm, respectively. Q_t = λ₀/Δλ represents the total quality of the side-coupled stub resonator (1/Q_t = 1/Q_o + 1/Q_e), where λ₀ and Δλ are the peak wavelength and the FWHM of the reflection spectrum. The intrinsic quality factor Q_o of the sub resonator can be estimated from [17]. So, in this structure, Q_o,j (j = 1,2) are about 500. Q_t,1 and Q_t,2 are about 63 and 61, respectively. Thus, Q_e,1 is about 72 and Q_e,2 is about 69.5. To further clarify the transmission characteristics, the transmission spectra versus δ and λ are shown in Fig. 5(c). Because of the finite mirror size in the structure, we set approximately δ = (d + 100) nm. It is found that the EIT-like transmittance varies periodically versus δ. For a determined δ, the transmission spectra agree well with that of Fig. 5(a). To exhibit more explicitly the formation and evolution mechanisms of plasmon-induced transparency by indirect coupling, we show in Fig. 5(d) the top view of Fig. 5(c). Dark (light) represents low (high) transmittance. As δ changes (resulting in the change of sin²θ in Eq. (6)), the transparency peak exhibits a shift and the symmetry of which also changes, while the transmission dips corresponding to resonant cavity modes remain unchanged. Consequently, for the case of plasmon-induced transparency by indirect coupling, the formation of transparency window is determined by the resonance detuning and the evolution of the transparency is mainly attributed to the change of coupling distance.

4. Conclusion

In summary, we have numerically and theoretically explored EIT-like spectral responses in plasmonic system composed of two sub resonators side-coupled to MDM waveguide. The results show that both the direct and indirect couplings between the two stub resonators lead to plasmon-induced transparency. For the directly coupled case, the EIT-like spectral response can be manipulated by adjusting the coupling distance and resonance detuning, which can be interpreted by the CMT. Decreasing the difference between the two stub resonators with fixed coupling distance gives rise to more pronounced slow-light effect. Moreover, the FWHM and group index of the transparent window to the inter-space are more sensitive than to the width of one cut, and the high group index of up to 60 can be achieved at visible frequency. For the case of only indirect coupling, the formation of transparency window is determined by the resonance detuning and the evolution of the transparency is mainly attributed to the change of coupling distance, which can be accurately analyzed by the FP model. Both the CMT and FP model agree well with the FDTD simulations. Hence, the plasmonic waveguide system may have potential applications for nanoscale optical switching, plasmonic sensing, and slow-light devices in highly integrated optical circuits.