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Abstract
We have theoretically and experimentally demonstrated electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) phenomena in a compact silicon ring-bus-ring-bus (RBRB) system. The two ring resonators in our RBRB system are both directly coupled and indirectly coupled through an asymmetric tricoupler. The coherent interference between a radiant mode and a subradiant mode in the two rings results in EIT and EIA effects at the through and drop ports, respectively. A theoretical model is established to analyze the proposed system based on temporal coupled mode theory. Finite-difference time-domain method is also employed to simulate the characteristics of this system. Consequently, RBRB structures were fabricated on a silicon-on-insulator platform and EIT and EIA transmissions have been observed simultaneously in the two outputs. The experimental results agree with our theoretical modeling and numerical simulations.
© 2017 Optical Society of America
1. Introduction
In the past decades, photonic analogue of electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) in atomic systems [1,2] has attracted significant attentions, as it can be realized much more easily and finds numerous applications in optical storage [3], slow light [4], optical switching [5], and nonlinear optics [6]. Though similar functions can also be accomplished by a single resonator, the EIT systems still have some special merits and advantages. For the dressed-state EIT systems, EIT resonances can be much narrower than those of individual resonators [7]. In addition, it is possible to reshape the input optical pulse, stop it and trap it, as the system is actively controlled [3]. For the bare-state EIT systems, EIT transmission is higher than that of individual resonator [8]. Furthermore, as the two resonances are properly detuned, this EIT system can exhibit Fano lineshape with greatly enhanced slope sensitivity [9], which is desirable for the realization of sensors and detectors. Until now, various EIT/EIA-like systems have been proposed and demonstrated, including directly coupled two resonators, such as series-cascaded microspheres/microtoroids [4,10], embedded microring resonators [6,11], and two-stage self-coupled optical waveguide resonators [12], and indirectly coupled two resonators, such as a parallel cascade of microrings [13–15], and photonic crystal cavities coupled to a waveguide [16,17]. Recently, EIT phenomena also have been observed in a single resonator where two whispering-gallery modes are indirectly coupled via an adjacent bus waveguide [18–24], and here the multimode resonator can be a microdisk [18,19], microsphere [20], microtoroid [21], bottle resonator [22], microbubble resonator [23], and quasicylindrical resonator [24]. However, it is still challenging to control the two resonant modes independently and freely in these systems, which makes the EIT effect not easy to get and thereby limits the applications, since the modifying or tuning of a waveguide-coupled resonator will change both two resonances simultaneously more or less. Hence, ring-bus-ring Mach-Zehnder interferometer (RBR-MZI) is proposed and fabricated to generate EIT [25,26], where the two resonant modes are coupled through a central waveguide and can be controlled independently. However, the usage of an additional MZI makes the whole structure much larger, which is undesirable for high-density on-chip photonic integration. A more simple structure, called dual microring resonators coupled via 3 × 3 couplers, is suggested in [27], but the performance of EIT and EIA is not so satisfying due to the usage of symmetric design of 3 × 3 couplers. Moreover, experimental work on this concept is still lacked so far.
In this paper, we have presented a ring-bus-ring-bus (RBRB) system where two ring resonators are coupled via an asymmetric 3 × 3 coupler (tricoupler) and demonstrated EIT and EIA effects in this structure on a nanophotonic silicon-on-insulator (SOI) chip experimentally. Using temporal coupled mode theory (T-CMT), we have established a theoretical model to describe the proposed system, with both direct and indirect coupling between the resonant modes taken into consideration, while only indirect coupling of resonant modes was considered previously [25]. Finite-difference time-domain (FDTD) simulations are also performed to study the EIT and EIA characteristics in this system. Then, silicon based RBRB systems were fabricated and characterized, and significant EIT and EIA phenomena appear at the through and drop ports, respectively. The experimental results, theoretical predictions, and numerical simulations show high consistency fundamentally. Owning to the design flexibility, simplicity, compactness, and good performance, we confirm that the on-chip EIT and EIA systems based on a RBRB structure are promising candidates for future applications in integrated photonics.
2. Modeling and simulations
As shown in Fig. 1(a), the presented RBRB system consists of a central waveguide sandwiched by two side-coupled ring resonators, one of which is coupled to an additional bus waveguide. In general, this resonators coupled system is achievable on various material platforms, and here we only take silicon-on-insulator for example, which it is compatible with integrated circuits and CMOS fabrication process. The structural parameters are labeled on Fig. 1(a). The widths of ring waveguide and bus waveguide are the same and denoted by W. R1 (R2) is the radius of ring 1 (ring 2), and g1 (g2) is the gap between ring 1 and the central bus (ring 2 and two buses). We input the light wave in the central bus, and observe the output lights at the through and drop ports. In this configuration, it is noteworthy that the two resonators are not only indirectly coupled via the tricoupler, but also directly coupled due to their proximity fields [28]. The coupling between the fields in the bus waveguides and two ring resonators can be illustrated by Fig. 1(b).
Fig. 1 (a) Schematic illustration of the RBRB system in silicon-on-insulator. (b) The coupling between fields in the waveguides and resonators.
To model and analyze the RBRB geometry, we employ the T-CMT [8,29,30] and have the following relations,
where represents the amplitudes of resonant modes in two ring resonators, Ω is a 2 × 2 Hermitian matrix with the diagonal elements representing the resonant frequencies (ω1 and ω2 for ring 1 and ring 2, respectively) and the nondiagonal elements being the direct coupling coefficient μ between the two resonant modes, and Γloss is a 2 × 2 Hermitian matrix standing for the decay rates of two resonant modes due to their intrinsic loss and energy escaping into the drop waveguide. We use 1/τe1 (1/τe2) as the decay rate due to energy in ring 1 (ring 2) escaping into the central bus (the central/drop bus), and 1/τo1 (1/τo2) as the decay rate due to intrinsic loss in ring 1 (ring 2). Thus, the matrices Ω and Γloss are given byIn Eq. (1), H is a 1 × 2 matrix describing the coupling between the central waveguide field and two resonant modes. Γport is a 2 × 2 Hermitian matrix denoting the decay rates to the central bus and indirect coupling between two modes through the central bus. Because of the energy conservation, we have Γport = H+·H/2 [8,29,30], where H+ is the Hermitian conjugate of H. Then, the expressions of H and Γport can be written as
Hence, we have obtained the following expressions,
According to Eq. (4) and the steady-state conditions, the transmission functions for the through (t = Sthr/Sin) and drop (d = Sdrop/Sin) ports can be derived as
where γ1 = iω-iω1-1/τo1-1/τe1, and γ2 = iω-iω2-1/τo2-2/τe2. Note that the RBRB geometry can be seen as a combination of a high-Q ring based all-pass filter and a low-Q ring based add-drop filter, and the destructive interference between high-Q (subradiant) and low-Q (radiant) resonant modes yields the desired bare-state EIT effect. Here, for comparison, the systems without ring 1 (i.e. add-drop filter) and without ring 2 (i.e. all-pass filter) are also calculated. The transmission functions of all-pass filter (t1) and add-drop filter (t2) are expressed asConsequently, the power transmissions T = |t|2, D = |d|2, T1 = |t1|2 and T2 = |t2|2 are simulated analytically and presented in Figs. 2(a)-2(d), as they share the same parameter values: ω2 = 1.21610 × 1015 rad/s (corresponding to λ2 = 1550.00 nm), 1/τo1 = 1/τo2 = 1 × 1010 rad/s, and 1/τe2 = 2 × 1011 rad/s. The direct coupling coefficient μ is assigned a negative value [31], and |μ| is set to be larger than 1/τe1 and lower than 1/τe2 in our case. When the high-Q ring 1 is in the over-coupling regime, significant EIT lineshapes are generated, as revealed in Figs. 2(a)-2(c). Note that the resonant wavelength of ring 1 should be slightly shorter than that of ring 2 to achieve an EIT response with two side dips of nearly equal depth, which is probably due to the resonance perturbation induced by the combination of individual resonators, and the EIT peak transmission is significantly higher than the dip transmission of ring 1. Moreover, the transmission and bandwidth of the EIT window decreases with 1/τe1, which can be engineered by changing the gap between ring 1 and central bus (g1). Figure 2(a) also shows that the EIA transmission is achieved at the drop port at the same time. It can be seen that the range of our EIT/EIA transmission is quite larger than that in [24], due to the use of an asymmetric tricoupler and great difference in the Q factors of two resonant modes. After that, we investigate the influence of neglecting the direct coupling term μ, which was implemented in previous work [25]. As illustrated in Fig. 2(d), to get a EIT response, the resonant wavelength of ring 1 should be the same as that of ring 2, and meanwhile the EIT peak transmission is just a little higher than the dip transmission of ring 1. The following FDTD simulations and experimental results will show us that the model considering direct coupling term is more accurate.
Fig. 2 (a) The transmission spectra of T, D, T1, and T2 with ω1 = 1.21623 × 1015 rad/s (λ1 = 1549.83 nm), 1/τe1 = 1 × 1011 rad/s, and μ = −1.5 × 1011 rad/s. (b) The transmission spectra of T, T1, and T2 with ω1 = 1.21617 × 1015 rad/s (λ1 = 1549.91 nm), 1/τe1 = 5 × 1010 rad/s, and μ = −1 × 1011 rad/s. (c) The transmission spectra of T, T1, and T2 with ω1 = 1.21614 × 1015 rad/s (λ1 = 1549.95 nm), 1/τe1 = 1 × 1010 rad/s, and μ = −5 × 1010 rad/s. (d) The transmission spectra of T, T1, and T2 with ω1 = 1.21610 × 1015 rad/s (λ1 = 1550.00 nm),1/τe1 = 1 × 1011 rad/s, and μ = 0.
Below, we will use two-dimensional FDTD method to numerically simulate the power transmission and field distributions of the proposed RBRB system. The incident wave is TE-polarized, and the calculation domain is surrounded by perfectly matched layer absorbing boundary. The waveguides are set with W = 0.3 μm, and the coupled ring resonators have R1 = 4.9981 μm, R2 = 5.0 μm, g1 = 0.15 μm, and g2 = 0.10 μm. The cladding material of waveguide is air (n = 1) and the core material is silicon, whose permittivity as a function of the wavelength is interpolated from the experimental results [32]. As seen in Fig. 3(a), EIT and EIA effects appear at the through and drop ports, respectively. The transmission spectra of ring 1 based all-pass filter and ring 2 based add-drop filter are also provided for comparison. The results show that the resonant wavelength of ring 1 is shorter than that of ring 2, and the EIT peak transmission is apparently higher than the dip transmission of ring 1. It is clear that the FDTD simulations are highly consistent with the theoretical modeling taking direct coupling into consideration, indicating the accuracy of the developed model for the RBRB system. Figure 3(b) shows the field distributions of Ez in the system at the EIT peak and left-dip wavelengths. We can find that the field is enhanced and mostly located in the high-Q ring 1, whereas the field in the low-Q ring 2 is suppressed and weak, when the system is on EIT resonance.
Fig. 3 (a) The transmission spectra of the RBRB system at the through and drop ports, the ring 1 based all-pass filter and ring 2 based add-drop filter at their through ports simulated by FDTD method. (b) Field distributions of Ez in the RBRB system at the EIT peak (point A) and left-dip (point B) wavelengths.
3. Experimental results
The proposed RBRB systems were fabricated on a SOI chip with a 220-nm-thick top silicon layer and a 2-μm-thick buried oxide layer. All the devices were patterned using electron beam lithography (EBL, Vistec EBPG 5000 plus). Then, the patterns on resist (ZEP520A) were transferred to silicon using inductively coupled plasma etching (ICP, Oxford Plasmalab System 100), and the etching depth is 220 nm. Silicon based RBRB structures have been realized, as the scanning electron microscope (SEM) image shows in Fig. 4(a). The cross-section size of all silicon waveguides is about 420 nm × 220 nm, ensuring the waveguides’ single-mode operation. The structural parameters of a RBRB system include R1, R2, g1, and g2, and we here only vary R1 and g1, while maintaining R2 = 5.00 μm and g2 = 0.11 μm to realize EIT and EIA effects in experiment for simplicity. Then, to characterize the silicon based RBRB systems, the integrated device and input/output fiber are coupled through a two-dimensional grating coupler for TE-polarization, which is shown in the left inset of Fig. 4(a). As indicated by the theoretical and numerical simulations, appropriate detuning of resonant wavelength/frequency between the subradiant and radiant modes is of critical importance. We know that the resonant wavelength of ring resonator can be adjusted by changing the radius. Aiming to quantify the relation, we fabricated ring based all-pass filters [see the inset of Fig. 4(b)] on the same chip of RBRB systems, with a radius of around 5 μm and a gap between the ring and bus of 0.15 μm. As shown in Fig. 4(b), the resonant wavelength increases generally with the radius. The error bars mainly reflect the influence of fabrication errors of waveguide dimensions.
Fig. 4 (a) SEM image of a SOI based RBRB device, insets: (left) two-dimensional grating coupler, (right) zoom-in of the asymmetric tricoupler (b) The relation between resonant wavelength and radius for a ring based all-pass filter.
Then, silicon RBRB systems with a footprint of ~22 μm × 11 μm and slightly changed R1 around 5 μm were fabricated and characterized by an optical transmission measurement when g1 = 0.15 μm, 0.17 μm, and 0.20 μm, as shown in Figs. 5(a)-5(c), respectively. It is seen that the spacing between the subradiant and radiant resonances changes with R1. As the two resonances are very close, the wide and narrow resonant dips are no longer distinct, and a narrow transparency peak occurs in the broad absorption band (i.e. EIT lineshape), which can be observed in the second row of Figs. 5(a)-5(c). From these curves, we find that the EIT peak transmission decreases and the Q factor of EIT resonance increases, when g1 increases. The measured peak transmissions are 0.95, 0.77, and 0.43, and the Q factors are about 2900, 5600, and 8700 for g1 = 0.15 μm, 0.17 μm, and 0.20 μm, respectively. The results can be explained by the fact that the decay rate of the subradiant mode’s energy escaping into the bus is reduced as the spacing gap (g1) is enlarged. The variation trend of EIT resonance agrees with the theoretical prediction previously. In addition, note that the EIT peak transmission is obviously higher than the dip transmission of ring 1, which becomes clear as the spacing between two resonances is sufficiently large. It is also found that the high-Q ring 1 is over-coupled for g1 = 0.15 μm and near critically-coupled for g1 = 0.20 μm. The experimental results are consistent with the theoretical and numerical simulations, validating the high accuracy of our analytical model for the RBRB system.
Fig. 5 The transmission spectra of RBRB systems with (a) g1 = 0.15 μm, (b) g1 = 0.17 μm, and (c) g1 = 0.20 μm.
The transmission spectra of the RBRB system with g1 = 0.17 μm and a proper R1 at two output ports are both presented in Fig. 6. It is seen that periodical EIT and EIA lineshapes are generated simultaneously at the through and drop ports, respectively, and they are generally complementary. We have used the model to fit the experimental results and extracted the key parameters, as shown in Fig. 6. The theoretical fitting exhibits good agreement with the experiment. The bandwidth and dip transmission of the left EIA valley are 0.27 nm and 0.06, respectively. The EIT and EIA spectra around 1549.3 nm are more symmetric and better than those around 1566.4 nm, since the free spectrum ranges of the subradiant and radiant modes are slightly different and the wavelength detuning between two resonances around 1549.3 nm is more appropriate. After all, as an EIT-like system, the proposed RBRB system is more flexible than single WGM resonator based systems [18–24] in engineering the two resonant modes separately, more simple and compact than a RBR-MZI based system [25,26] (footprint: ~145 μm × 24 μm) and a parallel cascade of two ring microrings (footprint: ~26 μm × 11 μm) [13,14]. It is also noteworthy that here we have experimentally demonstrated another configuration to achieve EIT and EIA in two output channels on a chip at the same time, which have only been realized in the parallel cascade of two microring resonators [13,14] until now to the best of our knowledge. In addition, the silicon RBRB system is potentially tunable, as the resonance positions can be independently controlled via electrical heating [33], nonlinear effect [34], and carrier injection [35]. Therefore, it is expected that the proposed RBRB geometry will find more applications in integrated optical interconnect and optical communications, because of the flexibility, simplicity, compactness, and good performance.
Fig. 6 The measured transmission spectra (solid lines) of the RBRB system with g1 = 0.17 μm and proper R1 at the through and drop ports. The dash-dot lines are theoretical fitting results, with ω1 = 1.21674 × 1015 rad/s (λ1 = 1549.19 nm), ω2 = 1.21664 × 1015 rad/s (λ2 = 1549.31 nm), 1/τo1 = 1/τo2 = 1 × 1010 rad/s, 1/τe1 = 3 × 1010 rad/s, 1/τe2 = 3 × 1011 rad/s, and μ = −2 × 1011 rad/s.
4. Conclusions
In summary, we have theoretically investigated and experimentally demonstrated photonic analog to EIT and EIA effects in a proposed RBRB system. The EIA and EIA lineshapes are generated by the coherent interference between a radiant mode and a subradiant mode in two ring resonators, which are both directly coupled and indirect coupled through an asymmetric tricoupler. We have developed a theoretical model to analyze the proposed structure using the T-CMT, and the FDTD simulation results show agreement with the analytical simulations. Using nanofabrication process, we fabricated RBRB structures on a SOI chip, and have observed significant EIT and EIA transmission spectra at the through and drop ports, respectively. The experimental results are consistent with the analytical and numerical simulation results, and validate the high accuracy of our theoretical model. The proposed EIT and EIA system is applicable in future integrated photonics circuits.
Funding
National Natural Science Foundation of China (NSFC) (61675084, 61335002, and 61435004); National High Technology Research and Development Program of China (2015AA016904).
Acknowledgments
The authors thank all the engineers in the Center of Micro-Fabrication and Characterization (CMCF) of Wuhan National Laboratory for Optoelectronics (WNLO) for the support in device fabrication. The authors also thank Prof. Yunfeng Xiao and Dr. Beibei Li in Peking University for valuable discussions.
References and links
1. I. Novikova, R. L. Walsworth, and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photonics Rev. 6(3), 333–353 (2012). [CrossRef]
2. M. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59(6), 4732–4735 (1999). [CrossRef]
3. Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3(6), 406–410 (2007). [CrossRef]
4. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef] [PubMed]
5. M. Mancinelli, P. Bettotti, J. M. Fedeli, and L. Pavesi, “Reconfigurable optical routers based on Coupled Resonator Induced Transparency resonances,” Opt. Express 20(21), 23856–23864 (2012). [CrossRef] [PubMed]
6. X. Zhou, L. Zhang, A. M. Armani, R. G. Beausoleil, A. E. Willner, and W. Pang, “Power enhancement and phase regimes in embedded microring resonators in analogy with electromagnetically induced transparency,” Opt. Express 21(17), 20179–20186 (2013). [CrossRef] [PubMed]
7. R. W. Boyd and D. J. Gauthier, “Photonics: Transparency on an optical chip,” Nature 441(7094), 701–702 (2006). [CrossRef] [PubMed]
8. Q. Lu, Z. Wang, Q. Huang, W. Jiang, Z. Wu, Y. Wang, and J. Xia, “Plasmon-Induced Transparency and High-Performance Slow Light in a Plasmonic Single-Mode and Two-Mode Resonators Coupled System,” J. Lightwave Technol. 35(9), 1710–1717 (2017). [CrossRef]
9. M. Tomita, K. Totsuka, R. Hanamura, and T. Matsumoto, “Tunable Fano interference effect in coupled microsphere resonator-induced transparency,” J. Opt. Soc. Am. B 26(4), 813–818 (2009). [CrossRef]
10. C. Zheng, X. Jiang, S. Hua, L. Chang, G. Li, H. Fan, and M. Xiao, “Controllable optical analog to electromagnetically induced transparency in coupled high-Q microtoroid cavities,” Opt. Express 20(16), 18319–18325 (2012). [CrossRef] [PubMed]
11. L. Zhang, M. Song, T. Wu, L. Zou, R. G. Beausoleil, and A. E. Willner, “Embedded ring resonators for microphotonic applications,” Opt. Lett. 33(17), 1978–1980 (2008). [CrossRef] [PubMed]
12. L. Zhou, T. Ye, and J. Chen, “Coherent interference induced transparency in self-coupled optical waveguide-based resonators,” Opt. Lett. 36(1), 13–15 (2011). [CrossRef] [PubMed]
13. Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006). [CrossRef] [PubMed]
14. S. Manipatruni, P. Dong, Q. Xu, and M. Lipson, “Tunable superluminal propagation on a silicon microchip,” Opt. Lett. 33(24), 2928–2930 (2008). [CrossRef] [PubMed]
15. A. Naweed, “Photonic coherence effects from dual-waveguide coupled pair of co-resonant microring resonators,” Opt. Express 23(10), 12573–12581 (2015). [CrossRef] [PubMed]
16. X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-Optical Analog to Electromagnetically Induced Transparency in Multiple Coupled Photonic Crystal Cavities,” Phys. Rev. Lett. 102(17), 173902 (2009). [CrossRef] [PubMed]
17. J. Pan, Y. Huo, S. Sandhu, N. Stuhrmann, M. L. Povinelli, J. S. Harris, M. M. Fejer, and S. Fan, “Tuning the coherent interaction in an on-chip photonic-crystal waveguide-resonator system,” Appl. Phys. Lett. 97(10), 101102 (2010). [CrossRef]
18. Q. Huang, Z. Shu, G. Song, J. Chen, J. Xia, and J. Yu, “Electromagnetically induced transparency-like effect in a two-bus waveguides coupled microdisk resonator,” Opt. Express 22(3), 3219–3227 (2014). [CrossRef] [PubMed]
19. M. Ghulinyan, F. R. Manzano, N. Prtljaga, M. Bernard, L. Pavesi, G. Pucker, and I. Carusotto, “Intermode reactive coupling induced by waveguide-resonator interaction,” Phys. Rev. A 90(5), 053811 (2014). [CrossRef]
20. C.-H. Dong, C.-L. Zou, Y.-F. Xiao, J.-M. Cui, Z.-F. Han, and G.-C. Guo, “Modified transmission spectrum induced by two-mode interference in a single silica microsphere,” J. Phys. B 42(21), 215401 (2009). [CrossRef]
21. Y.-F. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009). [CrossRef]
22. Y. Wang, K. Zhang, S. Zhou, Y.-H. Wu, M.-B. Chi, and P. Hao, “Coupled-mode induced transparency in a bottle whispering-gallery-mode resonator,” Opt. Lett. 41(8), 1825–1828 (2016). [CrossRef] [PubMed]
23. Y. Yang, S. Saurabh, J. Ward, and S. N. Chormaic, “Coupled-mode-induced transparency in aerostatically tuned microbubble whispering-gallery resonators,” Opt. Lett. 40(8), 1834–1837 (2015). [CrossRef] [PubMed]
24. X. Jin, Y. Dong, and K. Wang, “Stable controlling of electromagnetically induced transparency-like in a single quasi-cylindrical microresonator,” Opt. Express 24(26), 29773–29780 (2016). [CrossRef] [PubMed]
25. Y. Zhang, T. Mei, and D. H. Zhang, “Temporal coupled-mode theory of ring-bus-ring Mach-Zehnder interferometer,” Appl. Opt. 51(4), 504–508 (2012). [CrossRef] [PubMed]
26. S. Darmawan, L. Y. M. Tobing, and D. H. Zhang, “Experimental demonstration of coupled-resonator-induced-transparency in silicon-on-insulator based ring-bus-ring geometry,” Opt. Express 19(18), 17813–17819 (2011). [CrossRef] [PubMed]
27. X. Zhang, D. Huang, and X. Zhang, “Transmission characteristics of dual microring resonators coupled via 3x3 couplers,” Opt. Express 15(21), 13557–13573 (2007). [CrossRef] [PubMed]
28. M. S. Dahlem, “Studies of advanced integrated nano-photonic devices in silicon,” Ph.D. dissertation, Dept. Elect. Eng. Comput. Sci., Massachusetts Inst. Technol., Cambridge, MA, USA, 2011.
29. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multi-mode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]
30. M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, G. S. Wiederhecker, and N. C. Frateschi, “Modeling quasi-dark states with temporal coupled-mode theory,” Opt. Express 24(17), 18960–18972 (2016). [CrossRef] [PubMed]
31. Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066616 (2003). [CrossRef] [PubMed]
32. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).
33. G. Wang, A. Shen, C. Zhao, L. Yang, T. Dai, Y. Wang, Y. Li, X. Jiang, and J. Yang, “Fano-resonance-based ultra-high-resolution ratio-metric wavelength monitor on silicon,” Opt. Lett. 41(3), 544–547 (2016). [CrossRef] [PubMed]
34. M. Bernard, F. R. Manzano, L. Pavesi, G. Pucker, I. Carusotto, and M. Ghulinyan, “Complete crossing of Fano resonances in an optical microcavity via nonlinear tuning,” Photon. Res. 5(3), 168–175 (2017). [CrossRef]
35. Q. Huang, G. Song, J. Chen, Z. Shu, and J. Yu, “Proposal and Fabrication of an Electrooptically Controlled Multimode Microresonator for Continuous Fast-to-Slow Light Tuning,” IEEE Photonics J. 6(4), 2201011 (2014).
