We investigate electromagnetically induced transparency (EIT)-like effect in a metal–dielectric–metal (MDM) waveguide coupled to a single multimode stub resonator. Adjusting the geometrical parameters of the stub resonator, we can realize single or double plasmon-induced transparency (PIT) windows in the plasmonic structure. Moreover, the consistency between analytical results and finite difference time domain (FDTD) simulations reveals that the PIT results from the destructive interference between resonance modes in the stub resonator. Compared with previous EIT-like scheme based on MDM waveguide, the plasmonic system takes the advantages of easy fabrication and compactness. The results may open up avenues for the control of light in highly integrated optical circuits.
© 2014 Optical Society of America
Electromagnetically induced transparency (EIT) realized in atomic media originates from the quantum destructive inference between the excitation pathways to atomic upper level [1–3]. The EIT effect has potential applications for optical data storage and biosensor. However, the strict restrictions on the realization of the EIT hinder its practical application, which encourages an ongoing search for mimicking EIT in classical systems [4–24]. Surface plasmon polaritons (SPPs) trapped on the metal-dielectric interface can break the classical diffraction limit and manipulate light in the nanoscale domain [25–32]. Among various plasmonic guiding structures, MDM waveguides, which are easily fabricated and have deep subwavelength confinement of light with an acceptable propagation length for SPPs, have attracted much attention [5, 8, 14, 20, 21, 28, 32].
Based on the unique feature of MDM waveguide, the analog of EIT observed in the coupled optical resonator systems was theoretically predicted and experimentally demonstrated in recent research [5–9, 11, 13, 14, 20–23]. Zhang and associates introduced the EIT-like phenomena inside a plasmonic waveguide configuration with a comb line slot and rectangle cavity at infrared frequencies . Han et al. demonstrated the PIT using detuned Fabry-Perot resonators aperture-side-coupled to a MDM waveguide . A uniform theoretical model, for both direct and indirect couplings between the two stubs, was established to study PIT in the plasmonic stub waveguide , and Gong et al. realized on-chip PIT in a plasmonic waveguide side-coupled to a nanocavity pair . Recently, without employment of additional optical elements, EIT-like phenomena were demonstrated in a single plasmonic composite nanocavity [33, 34] or whispering-gallery microresonator [35–37]. However, there have been few studies on EIT-like in plasmonic MDM waveguide platform using a single stub resonator.
In this work, we numerically and analytically investigate the PIT spectral response in a plasmonic system composed of a MDM bus waveguide coupled to a single multimode stub resonator. The stub structure based on MDM waveguide was first reported by Matsuzaki et al. . It is well known that the stub has the advantages of small size and simple structure. Therefore, various components based on the stub structure have been demonstrated [7, 9, 13, 14, 20, 21]. Here, the configuration is also based on a single stub, but it exhibits different transmission characteristics as compared with . In particular, the proposed system is more easily fabricated than previously reported PIT structures [5–9, 11, 13, 14, 20–23, 33, 34], and we can observe single or double PIT windows in the plasmonic waveguide system. Therefore, this work may pave the way for the realization of nanoscale optical devices.
2. Analytical model
Figure 1(a) schematically shows MDM bus waveguide side-coupled to a stub resonator. The dielectric and metal in the structure are air and silver, respectively. The permittivity of silver is described by Drude model ε(ω) = 1 - ωp2/(ω2 + iωγp), with ωp = 1.38 × 1016 rad/s and γp = 2.73 × 1013 rad/s. These parameters are obtained by fitting the experimental results . ω is the angular frequency of incident light. The main structure parameters are the width (w) of bus waveguide, length (d) and width (L) of the stub resonator. As the TM polarization light is incident along x-axis, SPP wave can be formed on metal-insulator interface and confined in the waveguide. The width L of the stub resonator is large enough that the plasmonic stub resonator can provide multiple cavity modes. Using the quantum optics, we study a waveguide-coupled multimode system, as illustrated in Fig. 0.1(b). The Hamiltonian of the system can be written as [36, 37, 41]
The Heisenberg equations of motion for the waveguide and cavity modes can be given byEq. (2), we obtainEq. (4) into Eq. (3), we get
Applying, and the first Markov approximation, Eq. (5) can be simplified into20–22, 36, 37], we have
3. Simulation results and discussions
Figure 2(a) shows the transmission spectra from the same constructive parameters (w = 100nm, L = 300m), yet with d = 300nm, 340nm, 380nm, 400nm. It is obvious that a transparent window grows in strength and becomes more prominent as d increases from 300nm to 400nm. For the lossless metal case, the transmission spectra for d = 380nm (black dashed line) is also shown in Fig. 2(a). It can be seen that the transparent bandwidth keeps unchanged and we can obtain unity peak transmission in the transparent window.
The contour profiles of field Hz at different wavelengths are depicted in Figs. 2(c-f). Figure 2(c) corresponds to the resonant wavelength λ = 471.5nm for d = 300nm. Figures 2(d-f) illustrate z-component magnetic field, for the case of d = 380nm, at wavelengths λ = 528.3nm, 545.8nm, 556.2nm, respectively. The results in Figs. 2(c-f) are in conformity with the transmission spectra in Fig. 2(a). For the plasmonic stub structure, the resonance modes are denoted by TMmn, where m and n denote the number of node of standing waves in horizontal and vertical directions in the stub resonator. Using the definition of resonance mode, we aim to investigate the evolution of transmission spectra versus d. Figure 2(b) shows the resonant wavelengths and full width at half maximum (FWHM) of the transparent window in Fig. 2(a). Combining Fig. 2(a) and Fig. 2(b), we observe that both the wavelengths of resonance modes present red shift and the transparency window progressively widens with the increase of d.
As illustrated in Fig. 2(a), two resonance modes are simultaneously excited in the plasmonic stub system. In other words, the increase of d lifts the degeneracy of TM11 and TM02 modes. Based on the FDTD, we numerically extract the related physical parameters and obtain the theoretical results denoted by circles. It can be seen that the theoretical results, for the case of N = 2, are in good agreement with the FDTD simulations, which validates the correctness of the analytical model. So, the theoretical analysis allows us to understand the response of the plasmonic resonance system explicitly. From Eq. (7), we can see that the performance of the plasmonic system is dependent on the phase shift of coupling-out field κj1/2aj denoted by Cj. Consequently, to get insight into the physics of the EIT-like phenomena, we study, for d = 380nm in Fig. 2(g), the phase shifts of C1 and C2 corresponding to TM11 and TM02 mode, respectively. According to Eq. (6), we obtain that phase shifts of C1 and C2 are related to the input signal b+,in. Moreover, the interaction between C1 and C2 is determined by the relative phase shift of the two coupling-out fields and the phase difference between C1 and C2 is independent of the input signal b+,in. As shown in Fig. 2(g), the coupling-out field C1 nearly has a ~π phase difference with C2, which leads to the transparent response . Therefore, the EIT-like optical response results from destructive interference between TM11 and TM02 modes through the MDM bus waveguide, which is similar to the formation mechanisms of PIT in previously reported coupled-resonator structures [6, 8, 11, 14, 20].
Slow light is one of the most important applications for EIT effect in atom systems, and the plasmonic system also supports slow group velocities [8, 20, 34]. τg = dθ/dω and θ = arg(t) denote the optical delay time and transmission phase shift, respectively. The dispersion of transmission phase shift versus λ is plotted in Fig. 2(h). The normal dispersion, which leads to a significantly enhanced group delay, occurs in the transparent window, and the abnormal dispersion, which results in the negative group delay corresponding to the fast light (not discussed here), occurs around the plasmonic resonance modes. Combining Fig. 2(a) and Fig. 2(h), we can conclude that there is a trade-off between transmission ratio and the optical delay time, which agrees well with [7, 8, 20]. The EIT-like spectral response obtained by adjusting the length of the stub resonator may play an important role for the dynamic control of light in nanostructures. What impact does the width of stub resonator have on PIT effect?
In Fig. 3(a), we plot the transmission spectra as d = 300nm, w = 100nm, and L = 300, 280, 260, 220nm, respectively. Obviously, the FDTD simulations are in excellent agreement with the theoretical fittings. The transmission spectra exhibit typical EIT-like line shape with the decrease of L, and the transparency peak exhibits asymmetry, which is dependent on the coupling parameter κj and intrinsic-loss rate γj. The field distributions at mode TM11, transparency-resonance wavelength and mode TM02 are, respectively, depicted in Figs. 3(b-d). Both Figs. 3(b) and 3(d) show that the incident light is reflected as the local resonance is excited. In Fig. 3(c), the input light at the transparency-resonance wavelength can pass through the waveguide, which also originates from the destructive interference between modes TM11 and TM02. The resonant wavelengths of all modes versus L are plotted in Fig. 3(e). It is seen that TM11 mode presents red shift with the decrease of L, whereas the TM02 mode exhibits blue shift. The FWHM of the transparent window is also plotted in Fig. 3(e). Comparing Fig. 2(b) with Fig. 3(e), we find that the FWHM of transparent window to the width L of stub resonator is more sensitive than to the length of it, which offers a convenient scheme to control the bandwidth of transparent window. When L is larger than 300nm, whether the transmission spectra are similar to those in Fig. 3(a)?
Figure 4(a) illustrates the transmission spectra versus L. As shown in Fig. 4(a), the plasmonic system exhibits double EIT-like peaks when adjusting L from 360nm to 440nm. We note that the double EIT-like spectral response was only achieved in three-resonator system , highlighting our idea for the realization of double PIT windows in a single stub resonator. Figures 4(c-g) show z-component magnetic field, for the case of L = 420nm, at λ = 461.6nm, 482.9nm, 491.9nm, 498nm and 504.3nm, respectively. Figures 4(c), 4(e) and 4(g) show the profiles of field Hz of transmission dips corresponding to TM11, TM02 and TM20 modes, respectively. The resonant wavelengths of all modes versus L are plotted in Fig. 4(b). The resonant wavelength for TM11 (TM02) mode is in the range of 461.5-463.3nm (481.3-491.6nm), that is to say, they are nearly constant for L in the range of 360-440nm. However, in contrast to them, TM20 mode responds linearly to L. The fact indicates that the resonant modes are related to different formation and evolution mechanisms, and therefore it facilitates the dual-spectral PIT effects.
The transmission spectra calculated by the theoretical model is in good accordance with the FDTD method, from which the theoretical analysis also allows us to understand the dual-peak EIT-like response of the plasmonic resonator system. Consequently, in Fig. 4(h), we plot the phase shifts of coupling-out fields C1, C2 and C3 (corresponding to TM11, TM02 and TM20, respectively), for L = 420nm in Fig. 4(a), to analyze the dual-peak EIT-like response. In the left transparent window, C2 and C3 are in phase, but the coupling-out field C1 nearly has a ~π phase difference with C2 + C3, which leads to the transparent response. The other transparent peak, shown in Fig. 4(h), is attributed to the destructive interference between C3 and C1 + C2. The optical responses, for the cases of L = 360nm, 380nm, 400nm, and 440nm, can also be understood by studying the phase shifts of the coupling-out field. The dispersions of transmission phase shift are represented in Fig. 4(i), which indicates that normal dispersion occurs in the transparent windows. From the analytical model, the optical delay time in the transparent window is affected by the intrinsic loss in the stub resonator, which is consistent with [11, 42]. The transmission efficiency in the transparent window is associated with the phase shift of the coupling-out field, and is also affected by the intrinsic loss in the resonator. Thus, the incorporation of gain media in the plasmonic waveguide system compensates the intrinsic loss and enhances the EIT-like effect . As mentioned above, the single multimode stub scheme, which is more compact and easily fabricated than coupled resonator systems [5–9, 11, 13, 14, 20–23], may provide a possible way to realize the required EIT-like phenomena.
In summary, we have numerically and analytically demonstrated the PIT effect in a MDM bus waveguide coupled to a single multimode stub resonator. By manipulating the length or width of the stub resonator, we can obtain PIT windows in the plasmonic stub waveguide, and hence achieve the desired slow light effect. Moreover, the EIT-like spectral responses obtained theoretically are in good agreement with the rigorous FDTD simulations, which reveals that the PIT is attributed to the destructive interference between the resonance modes in the plasmonic stub resonator. In particular, the plasmonic system exhibits double transparency windows when adjusting L from 360nm to 440nm, and the double PIT response has only been shown in three-resonator system , highlighting our idea for the realization of double PIT windows in a single stub resonator. Owning to the simple configuration and compactness, the ultracompact structure holds great potential in sensitive sensor, optical switching and optical data storage in highly integrated optical circuits.
The authors are grateful to Prof. Yunfeng Xiao, Beibei Li and Yin Huang for their helpful discussions. This work was funded by the Fundamental Research Funds for the Central Universities of Central South University under Grant No. 2012zzts007 and 2013zzts009, the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20100162110068 and the National Natural Science Foundation of China under Grant No. 61275174.
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