An end-coupled composite-slot-cavity resonator (CSCR) is proposed based on the subwavelength metal-insulator-metal (MIM) waveguides. When compared with the perfect slot cavity, which acts as a Fabry–Pérot resonator, plasmonic-induced absorption effect is achieved in the CSCR system. Single or dual absorption windows will arise at the former transmission peaks by arranging the position of vertical-slot cavity in the CSCR. According to the analyses on the phase responses, abnormal dispersions will be achieved inside the windows. Therefore, one can manipulate the fast-light applications in the nano-scale integrated circuits. Furthermore, based on the same interference effect, plasmonic-induced transparency response with normal dispersion is also obtained by changing the end-coupled CSCR system to a side-coupled one. The performances of the proposed structure are analyzed and investigated using the coupled mode theory and the finite-difference time-domain method, respectively.
© 2017 Optical Society of America
Nowadays, increasing attention is paid to the development of integrated photonics, which is widely employed to solve the problems in optical communication areas [1–3], such as wavelength filtering and slow-light technology. Particularly for slow light, it is initially developed for the optical buffer and storage by using the fiber techniques. Recently, electromagnetically induced transparency (EIT) effect, which is first achieved in the laser-driven three-level atomic systems, is also suitable for the slow light owing to the normal dispersion in the transparency EIT window [4, 5], but the hard operation requirements limit its development. In this case, alternative plasmonic-induced transparency (PIT) effects with analogous-EIT characteristics are demonstrated by various metal-insulator-metal (MIM) waveguide structures [6–9], which have also been considered as one of the most promising ways to achieve the nano-integrated photonics circuits [10–14]. For example, PIT window is available in a side-coupled dual slot-cavity structure owing to the strong destructive interference effect between the dark mode and the bright mode [15–17]. In addition, PIT effects have also been observed in the dual/triple ring-resonator and the dual disk-resonator systems [18, 19], respectively. Therefore, on-chip slow-light applications, which are benefited from the large group delay time inside the PIT window, are widely developed by the subwavelength MIM structures. Distinguished from the transparency window, coherent absorption and dispersion engineering technologies [20–22], which have the advantages of perfect absorption window and easy dispersion management, are also widely studied. To obtain the comparable performances, plasmonic-induced absorption (PIA) effects, whose characteristics are completely contrary to that of the PIT system, have been demonstrated in the MIM structures . For example, PIA effect is observed in a compact MIM-based concentric nanoring resonator due to the mode interactions . Since abnormal dispersion will be available at the PIA window, it is believed that such MIM structures will be preferred for the on-chip ultra-fast light system [25, 26].
In this paper, an end-coupled composite-slot-cavity resonator (CSCR) based on MIM waveguides is proposed. A horizontal perfect slot cavity, which can be regarded as a Fabry–Pérot (FP) resonator, is placed between the input and output MIM waveguides. Owing to the mode interference, PIA effect will be available by adding a relatively vertical-slot cavity to one side of the end-coupled horizontal one. According to the 1st- and 2nd-order modes distributed inside the end-coupled slot cavity, single and dual PIA windows can be achieved by adjusting the position of the vertical cavity. Abnormal dispersions at the windows are also investigated, and thus it can find important applications in the on-chip fast-light area. By arranging this CSCR to be a side-coupled one, PIT response with normal dispersion can also be obtained due to the same interference effect. The performances of the proposed structure are analyzed and investigated by the coupled mode theory (CMT) and the finite-difference time-domain (FDTD) method, respectively.
2. Theory and simulations
Firstly, the end-coupled perfect-slot-cavity resonator (PSCR), which can be regarded as an FP resonator, is shown in Fig. 1(a). A horizontal-slot cavity is placed between the input and output MIM waveguides with coupling distance . According to the resonance condition, a series of longitudinal resonant modes will be oscillated in the cavity, and the wavelengths can be appropriately estimated through the phase condition:27]. By getting insight into the resonance mechanism, CMT method is employed to analyze the transmission spectrum [28, 29]. stands for the normalized amplitudes of SPPs in the output and input MIM waveguides, respectively, while is the one inside the PSCR. and are the quality factors related to the coupled loss from the PSCR into the MIM waveguide and the intrinsic loss inside the PSCR, respectively. Since SPPs are only launched into the MIM waveguide from the left side, the amplitude can be assumed as , and then the normalized amplitudes of the PSCR can be expressed as30]. The transmission spectrum is shown in Fig. 1(b) using black solid line. Obviously, there are two resonance peaks in the spectrum, i.e. the 1st-order mode at 1642 nm with a transmittance of ~0.39, and the 2nd-order mode at 840 nm with a transmittance of ~0.81. The resonance wavelengths agree well with that evaluated by Eq. (1). To further analyze the spectral response, CMT method is employed to plot the transmission spectrum. After setting and , the transmission spectrum, which is plotted in Fig. 2(b) with red-circle line, is in high accordance with that obtained by the FDTD method. The corresponding magnetic field distributions for the two modes are plotted in Fig. 1(c) and 1(d), respectively. There is a node and an anti-node for the 1st-order mode and the 2nd-order mode at the center of the PSCR, respectively, while both modes possess anti-nodes at the ends of the PSCR. Besides, two additional nodes for the 2nd-order mode arise at the quarter positions of the PSCR. Actually, the magnetic field distributions of resonant modes in the perfect or perturbed slot cavities have been theoretically and numerically analyzed [31–33]. The spectra are affected by the locations of the nodes and the antinodes, which we will use to design the PIA system.
Furthermore, a vertical-slot cavity, which can perform as an electromagnetic container, is placed above the PSCR to obtain the PIA response. This composite-slot-cavity resonator (CSCR), which consists of a horizontal-slot cavity and a vertical-slot one, is shown in Fig. 2. In view of these magnetic field distributions, single and dual PIA effects can be achieved by adjusting the position of the vertical cavity. Specifically, when the vertical-slot cavity locates at the anti-node of the expected mode inside the PSCR, PIA effect will arise at the wavelength of this mode. In this case, the shift from the center of the horizontal cavity is used to represent the position of the vertical cavity. To observe the PIA details by using CMT method, two new quality factors of and , which describe the coupled loss from the vertical cavity into the horizontal one and the intrinsic loss inside the vertical one, respectively, are added during the analysis. Then, the normalized amplitudes and of the horizontal- and vertical-slot cavities can be expressed as
Likewise, the transmission spectrum through the output MIM waveguide can be derived as
In the following, the length and width of the vertical-slot cavity are defined as and , respectively, the coupling distance between the two slot cavities is , and the other parameters show no changes during the simulations. Firstly, the vertical cavity is placed at the center of the horizontal one, i.e. , where is the anti-node for the 2nd-order mode and the node for the 1st-order mode in the horizontal cavity, respectively (see Fig. 1(c) and 1(d)). In this case, the 2nd resonance mode will be absorbed by the vertical cavity, while the 1st-order mode will not be affected. Figure 3(a) shows the transmission spectrum of the CSCR using black solid line based on the FDTD method. For convenience in comparison, the transmission spectrum of the PSCR is also plotted in Fig. 3(a). Obviously, PIA effect is successfully achieved in the CSCR according to the mode interactions, since the transmission peak at 840 nm (i.e. the 2nd resonance mode of the PSCR resonator) is replaced by a forbidden band. Meanwhile, the transmission peak for the 1st SPP mode remains at the same wavelength of 1642 nm. The results agree well with the above analyses. Besides, two new transmission peaks with high transmittances arise around the PIA window at the wavelengths of 769 nm and 924 nm, respectively. More details using the CMT method are illustrated in Fig. 3(b) with red-circle line. After setting and , PIA response is further investigated by the CMT method, which is highly in accordance with that using the FDTD method plotted using black solid line in Fig. 3(b). In general, the absorptivity A is calculated from its transmission T and reflection R by virtue of A = 1-R-T. To further investigate the absorption performance, the reflection and transmission spectra using FDTD method are also plotted in Fig. 3(c). Through the sum of the reflection and transmission, we can approximately obtain the absorption, which is up to 58.4%. Therefore, it is considered that PIA effect has been achieved in the CSCR system, although some of SPPs are reflected to the input MIM waveguide.
The variations of the phase response and the delay time with respect to the wavelength are plotted in Fig. 3(d) using blue dashed line and red solid line, respectively. Apparently, a phase shift occurs at the induced absorption window for the CSCR system. Since the phase and the delay time satisfy the condition: , it can be concluded that abnormal dispersion is achieved at the PIA window in view of the variation of the phase curve. From Fig. 3(d), we can see that ~-0.3 ps group time delay is obtained at the center wavelength of the absorption window. Besides, positive time delays of ~0.1 ps and ~0.05 ps are also available in the left and right transmission peaks, respectively, and thus, a slight normal dispersion will be obtained by using the two peaks. As for the 1st-order SPP mode at 1642 nm, the phase curve becomes smoother than that within the band gap, leading to a quite little delay time of ~0.02 ps. Subsequently, PIA response is theoretically confirmed and one can develop the on-chip ultra-fast-light applications by using this kind of CSCR system.
The magnetic field distributions for the forbidden band and the transmission peaks, which are marked with “I, II, III” in Fig. 3(a), are shown in Figs. 3(e)-3(g). Obviously, SPPs for the band gap will be stopped by the CSCR while the ones for the transmission peaks will propagate through the output MIM waveguide. Interestingly, the reverse phase of the magnetic field in the horizontal-slot cavity of Fig. 3(f) (i.e. corresponding to the PIA window) is obtained in contrast to that in Figs. 3(e) and 3(g). Destructive interference will then arise and a band gap with ultra-low transmission will occur at the former transmission peak.
The lengths of the horizontal- and vertical-slot cavities are also changed to investigate their influences on the PIA response, respectively. In Figs. 4(a)-4(e), the length of the horizontal cavity increases from 440 nm to 600 nm with a step of 40nm, while the length of the vertical one remains the same as 490 nm. The transmission spectra are plotted by using blue dashed line, and it can be seen that both transmission peaks have linear redshifts by increasing . The PIA effect can be always obtained because an absorption band is achieved in Figs. 4(a)-4(e). The results are further confirmed by the curves of the time delay (red solid line), since normal and abnormal dispersions are obtained at the peaks and the dips, respectively. However, the time delay also investigates that the response wavelength for the PIA window will not be changed by . In contrast, is fixed as 520 nm and increases from 410 nm to 570 nm with a step of 40 nm in Figs. 4(f)-4(j). In this case, the two transmission peaks and the PIA window have linear redshifts, which is further demonstrated by the time delay. Thus, it is believed that the wavelength of the PIA window is only affected by , but the two transmission peaks are changed by both and .
Then, the vertical-slot cavity is moved to the position of , where is almost the anti-node of the 1st-order mode at 1642 nm and the node of the 2nd-order mode at 840 nm, respectively. In this case, only the SPPs for the 1st-order mode in the horizontal-slot cavity will be captured into the vertical one, leading to the destructive interferences. The transmission spectra of the PSCR and CSCR systems are shown in Fig. 5(a) based on FDTD simulations. When compared with the transmission spectrum of the PSCR system with red dashed line, the CSCR system also has a transmission peak plotted using black solid line at 840 nm. However, the former transmission peak at 1642 nm is replaced by an absorption window in the CSCR system, and two additional peaks arise at 1527 and 1789 nm, respectively. The results, which are further confirmed by using CMT method using red circle line, are in high accordance with that by using FDTD simulation in Fig. 5(b). Figure 5(c) shows the phase variation and the group time delay, and it can be observed that the obvious phase shifts occur at the PIA window, leading to ~-0.13 ps time delay at 1642 nm. Thus, it is believed that a significant abnormal dispersion is obtained, and one can manipulate fast light by using this window. Meanwhile, normal dispersions are available for those two peaks, which are also investigated by the responses of the group time delay. The magnetic field distributions for the PIA window and the two peaks marked with “I, II, and III” are shown in Figs. 5(d)-5(f), respectively. SPPs can pass through the output waveguide in Figs. 5(d) and 5(f), but the one in Fig. 5(e) is stopped by the resonator.
Interestingly, the anti-nodes for the 1st and 2nd modes coexist at the end of the horizontal-slot cavity. When the vertical-slot cavity is moved to this position, as shown in Fig. 6(a), both modes will be captured into the vertical cavity from the horizontal one. Because of the mode interactions, dual absorption windows are achieved at 1642 nm and 840 nm for the CSCR system in Fig. 6(a) using black solid line. The PIA effects are further investigated by the phase responses and the group delay time in Fig. 6(b). About −0.30 ps and −0.12 ps time delays are obtained at the center wavelengths of the two windows, respectively, and thus it is considered that abnormal dispersions are available inside the two windows. The magnetic field distributions for the transmission peaks and windows marked with “I, II, III, IV, V, and VI” are shown in Figs. 6(c)-6(h), which illustrate the propagation details of the SPP modes. In Figs. 6(d) and 6(g), SPPs at the center wavelengths of the windows will be stopped due to the destructive interferences in the slot cavities.
In view of the similar interference effect, when the end-coupled CSCR system is manipulated to be a side-coupled one in Fig. 7(a), PIT performance should be achieved and normal dispersion can be obtained in the transparency window. According to the similar process using CMT method, the transmission of the side-coupled CSCR can be written as:
Figure 7(b) shows the transmission spectrum and the phase response of the side-coupled PSCR structure. The 1st-order and the 2nd-order resonance modes arise at the wavelengths of 1640 nm and 842 nm in Fig. 7(b), respectively. After adding a vertical-slot cavity to the side-coupled PSCR system, single or dual PIT effects occur in Fig. 7(c)-7(e) by arranging the position of the vertical-slot cavity. First, a transparency window with high transmittance is obtained at 842 nm with in Fig. 7(c), but the forbidden band at 1640 nm shows no change. Meanwhile, a phase shift is also available within the transparency window. After changing the vertical-slot cavity to the position of in Fig. 7(d), the transparency window is transferred to the wavelength of the 1st-order mode at 1640 nm. Finally, dual PIT effects are achieved in Fig. 7(e) with . From the above analysis, this position is the anti-node for both resonance modes. Because of mode interactions, the two forbidden bands at 842 nm and 1640 nm are replaced by dual transparency windows, respectively. Therefore, on comparing with the PIA effect in the end-coupled CSCR system, the corresponding PIT effect is also investigated in the side-coupled CSCR system.
In conclusion, plasmonic-induced absorption has been analyzed and investigated in an end-coupled CSCR structure by using the CMT theory and the FDTD method, respectively. When compared with the band-passed filtering PSCR structure, single and dual absorption windows arose at the former transmission peaks by arranging the position of the vertical-slot cavity. Since −0.3 ps delay time has been investigated inside the window, this kind of structure would be preferred in the on-chip ultrafast light applications due to the abnormal dispersion. It was also demonstrated that the center wavelength at which the PIA effect occurred was only affected by the length of the vertical cavity. Besides, PIT effect has been achieved by changing the end-coupled CSCR system to be side-coupled one due to the same interference effect. Slow-light application could also be developed by using the proposed structure.
National Natural Science Foundation of China (61405039, 61475037); Science and Technology Planning Projects of Guangdong Province, China (2016A020223013); the Natural Science Foundation of Guangdong Province, China, (2014A030310300); the State Key Lab of Optical Technologies for Micro-Engineering and Nano-Fabrication of China; the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2014KQNCX066); and the Research Fund of Guangdong University of Technology (16ZK0041, 13ZK0387).
The authors would like to thank the editor and the reviewers for their valuable comments and suggestions, which help improve the quality of the manuscript.
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