Open Access
Add to BibSonomy
Abstract
Waveguide crossing is an important integrated photonic component that will be routinely used for high-density and large-scale photonic integrated circuits, such as optical switches and routers. Several techniques have been reported in achieving high performance waveguide crossings on a silicon-on-insulator photonic platform, i.e., low-loss and low-crosstalk waveguide crossings based on multimode interference, bi-layer tapering, optical transformation, metamaterials, and subwavelength gratings. Until recently, not much attention has been given to the reduction of the footprint of waveguide crossings. Here we experimentally demonstrate an ultra-compact waveguide crossing on silicon photonic platform with a footprint only ~1 × 1 μm2. Our simulations show that it has a low insertion loss (< 0.175 dB) and low crosstalk (< −37dB) across the whole C-band, while the fabricated one has an insertion loss < 0.28 dB and crosstalk around −30 dB for the C-band.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Increasing the integration density for electronic integrated circuits (IC) has been the driving force behind modern semiconductor industry. Moore's law states that the integration density of transistors doubles almost every 18 months. In recent years, there are signs of technological difficulties in keeping up with this law. The rapid development of emerging CMOS photonics is believed to be able to alleviate such difficulties.
In the past decades, various photonic functionalities have been demonstrated on CMOS photonic platform. Although, attentions have not been fully focused on the miniaturization of photonic components so as to achieve high-density photonic integration, there are definitely more and more research studies on demonstrating ultra-compact silicon photonic devices. For instances, 1) an integrated silicon-based wavelength demultiplexer with a footprint of 2.8 × 2.8 μm2 was designed and implemented by Piggott et al. [1], 2) an ultra-compact hybrid plasmon/silicon nanowire asymmetric direct coupler was proposed with a device footprint of 1.9 × 3.7 μm2 [2], 3) the design and experimental demonstration of an on-chip polarization beam splitters (PBS) with a device area of only 2.4 × 2.4 μm2 was reported by Shen et al. [3], and 4) a multi-channel all-optical switch was demonstrated by Dong et al. with the size of each unit being only 8 × 3.9 μm2 to meet the growing demand for compact optical switches for photonic integrated circuits [4]. Clearly, moving towards such a direction will be favored economically because the reduction of device footprint means a lowered manufacture cost, and it also represents a fundamental interest for scientists to discover new principles that allow for designing unconventional ultra-compact integrated devices. The silicon-on-insulator (SOI) platform has a high refractive index contrast, enabling tight optical confinement with a typical waveguide dimension about hundreds of nanometers [5]. Although this tight confinement is very beneficial for designing compact devices, i.e., those mentioned above, it is not favored for certain applications. For instance, a waveguide crossing is a device connecting two bus waveguides with a high transmission in the straight path and a low crosstalk to the crossing path. Its simplest form, i.e., the crossing of two perpendicular bus waveguides, which usually performs well on SiO2 platform, will fail to work if no careful modification is added to the crossing section on SOI platform. This is due to the tight confinement or more accurately speaking its resulted large divergence angle in the waveguide crossing section. As more photonic functionalities, e.g., optical switching and routing, are integrated on chip, the presence of the waveguide crossing seems inevitable. Several possible waveguide-crossing structures on SOI platform have been reported. For example, the cross angle of the two bus waveguides may be optimized [6]. But the improvement of overall performance is very limited. Another design that can offer better performance is to utilize the self-imaging property in multimode-interference structures [7]. However, its large 13 × 13 μm2 footprint may be an issue for dense integration. Recently there have been reports attempting to further reduce the footprint [8–11] with successful demonstrations with a 6 × 6 μm2 footprint [9,10]. Note that fabricating the demonstrated structure in [10] needs an accurately aligned two-step etching process. A similar structure utilizing only a simple one-step etching process with comparable performance and footprint was reported in [8,9,11]. Lately, a new kind of high efficient waveguide-crossing structure was proposed in [12] and experimentally demonstrated in [13], where a second waveguide on top of two silicon inverse tapers was used as a crossing bridge. There are also other types of waveguide-crossing structures based on principles, e.g., Bloch-wave in a periodic array [14] and sub-wavelength gratings [15]. All the waveguide-crossing structures mentioned above rely on single mode operation. The rising of on-chip spatial division multiplexing technology has demanded the waveguide crossings operating in the multi-mode domain [16,17]. Nevertheless, the footprint for these waveguide-crossing structures remains relatively large, which may have issues for dense integration.
In this study, we demonstrate a new kind of ultra-compact and highly efficient waveguide-crossing structure on SOI platform with a footprint as small as ~1 × 1 μm2. Within the C-band operation, our waveguide crossing is very efficient with simulations showing the insertion loss less than 0.175 dB and the crosstalk below −37 dB and with experimental test results showing the insertion loss < 0.28 dB and the crosstalk around −30 dB. Our device may find potential applications when the design constraints are slightly geared towards device footprint [18,19], e.g., high-port-count optical switching fabrics [20], where a large number of waveguide crossings are needed. This forward-looking design may have great impact and practicality when the most up-to-date fabrication CMOS capability is adopted in this field.
Recently, the inverse design approach for ultra-compact photonic devices has attracted much attention [21–26] due to its object-oriented design principle. We leverage a similar concept and apply it here to design ultra-compact waveguide crossings. First, the structure is constructed on the basis of a typical waveguide crossing design that consists of two identical air cladded SOI waveguides (width × height = 500 nm × 220 nm) and linear tapers surrounding the waveguide intersection as shown in Figs. 1(a) and 1(b). Next, fully etched holes are added to the design region outlined by the yellow box, which contains the whole waveguide-crossing structure. Finally, their exact positions and sizes along with other key device geometric parameters are randomly searched by an optimization algorithm with the goal to find a device with the best performance. Note that the placement of etched holes and tapered waveguides has to satisfy a 4-fold rotation symmetry and horizontal, vertical and diagonal mirror symmetries. Thus, the resulted basic repeating unit is a triangular section denoted by the blue triangle.
Fig. 1 (a) Schematic of our waveguide-crossing structure, (b) zoom-in view of one tapered section, and (c) zoom-in view of the etched holes. Yellow box denotes the design region, and the blue triangle denotes the basic repeating unit of the design region.
In our optimization, electromagnetic simulations for the waveguide crossing are conducted with a 3D FDTD engine using a fine grid size down to 6 nm. This engine is driven by a particle swarm optimization (PSO) algorithm [11,27]. Detailed parameters to be optimized include linear taper parameters C1 and C2 as shown in the Fig. 1(b), and the number of etch holes in the basic repeating unit i, their corresponding locations Xj and Yj and their radii rj (j = 1,2,..,i) as shown in Fig. 1(c). Our design goal is to obtain a waveguide crossing with high fundamental quasi-TE mode transmission for C-band telecommunication. This translates into an optimization figure of merit (FOM), which is the averaged wavelength-dependent transmission in the C-band. The number of particles and iterations are set to 20 and 500, respectively. Sellmeier equations are used in simulation to incorporate the material dispersions for silicon and silicon dioxide [28]. The parameters for an optimized waveguide-crossing structure are summarized in Table 1, and the corresponding schematic is illustrated in Fig. 1(a). Notice that all the optimized structures always go through a grid-size convergence study before their final performance data are given in this work. This device allows for a very low averaged insertion loss ~0.22 dB or high averaged transmission > 95% across the entire simulation wavelength window. More importantly, the footprint is very small, only about ~1 × 1 μm2, almost two orders of magnitude smaller than those reported previously [8–11].
Table 1. Waveguide-crossing structure parameters (μm)
In order to check the consistence of our optimization results, three more cases are studied, where the number of etched holes inside the basic repeating unit are fixed to be 3, 4 and 5, as opposed to 2, a final optimized value from the previous case. The corresponding optimized designs with transmission >95% for each case after 100 iterations are shown in Fig. 2. It is clear that regardless of the number of holes allowed in the optimization the final designs all contain holes inside the crossing and most importantly the shapes formed collectively by these holes are very similar. In fact, they resemble closely etched lens-like structures. Inspired by this observation, we redesign the waveguide-crossing structure by replacing the holes with the lens-like structures as indicated in Fig. 3(a), and perform the PSO again but with a new set of geometric parameters for the lens-like structures including their offset (P) (defined from the center of the waveguide crossing), thickness (W), length (D), front radius of curvature (R1) and back radius of curvature (R2). The final optimized parameters are summarized in Table 2. Notice that the size of the lens-like structures is less than 100nm, which makes fabrication process very challenging with current CMOS photonics technology. However, as the CMOS photonics manufacturing technology evolves and adopts much smaller node size in the foreseeable future, such difficulties will be no longer a limiting factor.
Fig. 2 Schematics for the optimized waveguide crossing design with 3, 4 and 5 etch holes in the basic repeating unit, respectively.
Fig. 3 (a) Schematic of our waveguide-crossing structure with etched lens-like structures, (b) zoom-in view of one tapered section, and (c) zoom-in view of the etched lens-like structures.
Table 2. Waveguide-crossing structure parameters (μm)
2. Device performance
The simulated performance of our waveguide crossing in Fig. 3 is plotted in Fig. 4. Its insertion loss has a parabolic shape with a maximum less than 0.175 dB or larger than 96% transmission within the C-band. Furthermore, as shown in Fig. 4, our device has a very low crosstalk, < −37 dB for the C-band with a distinctive dip (about −48dB) at around 1550 nm. The emergence of such a dip is a result of destructive interference of the reflected waves from the two lens-like structures in the crossing path. The spectral location of this dip coincides more or less with that of the minimum insertion loss as shown in Fig. 4. Such a unique spectral response can be explained by the fact that the two lens-like structures in the transmission path act as two partial reflecting mirrors and form a Fabry-Perot (FP) cavity. Consequently, the low insertion loss and high transmission of our device are enabled by the resonance tunneling from this FP cavity. Notice that another resonance wavelength for this structure can be found using the FP cavity theory and discovered by our FDTD simulation to be around 940 nm. This confirms that the validity of our explanation for the principle behind this waveguide-crossing structure. Further FDTD simulations also reveal that a substantial portion of the light, more than 3%, is leaked into the substrate, which in principle could be largely suppressed by removing the substrate underneath the waveguide crossing and making it suspended in air [29]. When this new structure is combined with even smaller features allowed in design, waveguide-crossing structures with better performance could be confidently achieved. Nevertheless, the above results suggest that our device has very competitive performance but with a much smaller footprint, ~1 × 1 μm2, compared with those reported in [6–15].
The mechanism behind the high transmission of our device can be further revealed by examining the power flow under different scenarios as shown in Fig. 5. In all these plots, the fundamental quasi-TE mode at 1550 nm is injected from the bottom of the vertical waveguide. As shown in Fig. 5(a), when all four etched lens-like structures are present (corresponding to the actual optimized case), there is negligible diffraction into the crossing waveguide and no interference pattern observed in the injection waveguide, indicating a very low crosstalk and back reflection. When the two lens-like structures in vertical direction are removed as shown in Fig. 5(b), the rest two lens-like structures in horizontal direction can inhibit diffraction by forming a waveguide guiding region in the crossing section and thus allowing a high power transmission through the crossing. When only the two lens-like structures along horizonal direction are removed as illustrated in Fig. 5(c), the device performance degrades with clear indications of power leak into the crossing waveguide (strong crosstalk) and interference pattern in the injection waveguide (strong back reflection). The above findings imply that the working principle behind our device is fundamentally different from the self-imaging effect due to the multi-mode interference in waveguide crossing reported in [8,9,11], where the center portion does not provide any index guiding but instead is a free-propagation region.
Fig. 5 Steady state power flow for our waveguide-crossing structure, (a) with all four lens-like structures, (b) with only two lens-like structures in horizontal direction, and (c) with only two lens-like structures in vertical direction. Solid lines denote the outline of the waveguide-crossing structure.
Fabrication tolerance could be an important factor influencing the performance of our proposed waveguide-crossing structures. It is thus numerically investigated using the PSO algorithm. However, in this scenario, PSO algorithm tries to find the case with the worst possible fundamental quasi-TE mode transmission, when the design parameters (i.e, C1, C2, D, W, P, R1 and R2 shown in Fig. 3) are allowed to vary within [0, δ], where δ is a value from −10 nm to 10 nm. The obtained result for the wavelength at 1550 nm is summarized in Fig. 6. Clearly, the performance of our device is hardly influenced by the small variations of all main geometric parameters. It only shows slight degeneration when the variation is more than + 5 nm. But the crosstalk performs quite well, < −32.8 dB, even for ± 10 nm deviation.
3. Fabrication and characterization
The waveguide crossings designed above are fabricated on a 220 nm SOI wafer with a 2 μm buried dioxide buffer layer using electron beam lithography and inductively coupled plasma (ICP) etching. Devices with an increased number of crossings (i.e., 5, 10, 15, 20, 25 and 30 crossings) are fabricated in order to characterize the performance of our proposed waveguide-crossing structures using cut-back measurement method. Scanning electron micrograph (SEM) images of a typical device are shown in Fig. 7. Several waveguide crossings are arranged in the transmission path while the first one in the path is used to measure the crosstalk. The inset of Fig. 7 shows the details of our crossing structure, where the four deep sub-wavelength lens-like structures can be clearly identified. It should be pointed out that, although a dedicated overlay patterning step has already been used to pattern these extreme small structures in order to alleviate the large dose contrast issue between them and the waveguide region, a careful statistical analysis on them reveals that these structures have been slightly over-exposed by roughly 6.4 nm and a straightforward visual examination of SEM pictures suggests that their shapes are slightly distorted as well. Nevertheless, the corresponding performance, characterized by a broadband light source and an optical spectrum analyzer of these devices, is still very reasonable. In detail, coupling in and out of the devices is realized via a pair of grating couplers. The measured insertion loss spectra (including coupling loss) for devices with 5, 10, 15, 20, 25 and 30 waveguide crossings along with a reference waveguide without any crossings are shown in Fig. 8(a). By subtracting the insertion loss spectrum of the reference waveguide from those of the devices with cascaded crossings, the insertion loss spectra for the cascaded crossings alone can be obtained. By linearly fitting the insertion loss data for 5, 10, 15, 20, 25 and 30 crossings at a single wavelength, the insertion loss per waveguide crossing at this wavelength can be extracted from the slope of the fitted line. By repeating this procedure over the whole wavelength range (from 1528 to 1567 nm) with a wavelength resolution of 0.1 nm, the whole spectrum of wavelength-dependent insertion loss per crossing is obtained and plotted in Fig. 8(b). It can be seen that our fabricated device has an insertion loss less than 0.28 dB within the C-band and the insertion loss curve is relatively flat, i.e., about 0.25 dB for longer wavelengths. Compared with our simulation results, the performance of this device is only slightly degraded, which can be attributed to an increased scattering loss as a result of fabrication imperfection discussed above. The crosstalk of our devices is reasonably low, with a measured value around −30 dB across the whole spectral window of interests as shown in Fig. 8(b). The ripples on the crosstalk are due to the interference of the multiple scattered light from not only the first waveguide crossing but only all the test of waveguide crossings on the test device. Similar effect is commonly observed in measurements reported elsewhere [30–32]. Since the crosstalk level in our case is very low, these ripples can be safely neglected for most applications.
Fig. 7 SEM images of a typical fabricated waveguide crossing device. The inset shows the details of the center crossing region.
Fig. 8 (a) Insertion loss for devices with a different number of waveguide crossings. (b) Insertion loss and crosstalk for a single typical waveguide-crossing structure over the range from 1528 nm to 1567 nm.
4. Conclusion
By leveraging an inverse design concept, we are able to successfully design ultra-compact waveguide-crossing structures that are completely different from those reported elsewhere in literatures. We show that a highly efficient and ultra-compact waveguide-crossing structure based on linear tapers and a pattern of etched holes can be designed regardless of the number of holes used in the optimization. The fact that the shapes formed by these holes are more or less the same leads to a more elegant and meaningful design using etched lens-like structures. The footprint of both designs can be more than 10-fold smaller than that of previous reported structures. Detailed numerical analysis shows that our device with etched lens-like structures has an insertion loss less than 0.175 dB, and crosstalk less than −37 dB within C-band. Despite the fabrication imperfection, our device is still robust with measured insertion losses less than 0.28 dB and crosstalk around −30 dB. Its performance can in principle be further improved with a suspended design and better fabrication processes. This new kind of forward-looking waveguide crossing design holds great promises in high-density integration of silicon photonic circuits and its potentials could be fully utilized soon after the state-of-the-art CMOS fabrication technology becomes the mainstream in the field of silicon photonics.
Funding
National Basic Research Program of China (Grant 2015CB659400); Natural Science Foundation of Jiangsu Province (Grant BK20150057); Fundamental Research Funds for the Central Universities (Grant 021314380100).
References
1. A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015). [CrossRef]
2. X. Guan, H. Wu, Y. Shi, L. Wosinski, and D. Dai, “Ultracompact and broadband polarization beam splitter utilizing the evanescent coupling between a hybrid plasmonic waveguide and a silicon nanowire,” Opt. Lett. 38(16), 3005–3008 (2013). [CrossRef] [PubMed]
3. B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9(6), 378–382 (2015). [CrossRef]
4. G. Dong, W. Deng, and X. Zhang, “Fast and Ultra-compact Multi-channel All-optical Switches Based on Silicon Photonic Crystal Nanobeam Cavities,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM4B.5. [CrossRef]
5. M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nat. Photonics 4(8), 492–494 (2010). [CrossRef]
6. P. Sanchis, J. Galan, A. Griol, J. Marti, M. Piqueras, and J. Perdigues, “Low-crosstalk in silicon-on-insulator waveguide crossings with optimized-angle,” IEEE Photonics Technol. Lett. 19(20), 1583–1585 (2007). [CrossRef]
7. H. Chen and A. W. Poon, “Low-loss multimode-interference-based crossings for silicon wire waveguides,” IEEE Photonics Technol. Lett. 18(21), 2260–2262 (2006). [CrossRef]
8. Y. Zhang, S. Yang, E. Lim, G. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss, and low-crosstalk silicon waveguide crossing,” IEEE Photonics Technol. Lett. 25(5), 422–425 (2013). [CrossRef]
9. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. 34(18), 2760–2762 (2009). [CrossRef] [PubMed]
10. W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Opt. Lett. 32(19), 2801–2803 (2007). [CrossRef] [PubMed]
11. Y. Ma, Y. Zhang, S. Yang, A. Novack, R. Ding, A. E.-J. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect,” Opt. Express 21(24), 29374–29382 (2013). [CrossRef] [PubMed]
12. A. V. Tsarev, “Efficient silicon wire waveguide crossing with negligible loss and crosstalk,” Opt. Express 19(15), 13732–13737 (2011). [CrossRef] [PubMed]
13. A. M. Jones, C. T. DeRose, A. L. Lentine, D. C. Trotter, A. L. Starbuck, and R. A. Norwood, “Ultra-low crosstalk, CMOS compatible waveguide crossings for densely integrated photonic interconnection networks,” Opt. Express 21(10), 12002–12013 (2013). [CrossRef] [PubMed]
14. M. Popovic, E. P. Ippen, and F. Kartner, “Low-loss bloch waves in open structures and highly compact, efficient Si waveguide-crossing arrays,” in Lasers Electro-Optics Soc. 20th Annu. Meet. (IEEE. 2007), pp. 56–57. [CrossRef]
15. P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, D.-X. Xu, S. Janz, A. Densmore, and T. J. Hall, “Subwavelength grating crossings for silicon wire waveguides,” Opt. Express 18(15), 16146–16155 (2010). [CrossRef] [PubMed]
16. C. Sun, Y. Yu, and X. Zhang, “Ultra-compact waveguide crossing for a mode-division multiplexing optical network,” Opt. Lett. 42(23), 4913–4916 (2017). [CrossRef] [PubMed]
17. H. Xu and Y. Shi, “Dual-mode waveguide crossing utilizing taper-assisted multimode-interference couplers,” Opt. Lett. 41(22), 5381–5384 (2016). [CrossRef] [PubMed]
18. J. Wang, C. Lee, B. Niu, H. Huang, Y. Li, M. Li, X. Chen, Z. Sheng, A. Wu, W. Li, X. Wang, S. Zou, F. Gan, and M. Qi, “A silicon-on-insulator polarization diversity scheme in the mid-infrared,” Opt. Express 23(11), 15029–15037 (2015). [CrossRef] [PubMed]
19. H. Subbaraman, X. Xu, A. Hosseini, X. Zhang, Y. Zhang, D. Kwong, and R. T. Chen, “Recent advances in silicon-based passive and active optical interconnects,” Opt. Express 23(3), 2487–2510 (2015). [CrossRef] [PubMed]
20. J. Lu and J. Vučković, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013). [CrossRef] [PubMed]
21. P. Wahl, T. Tanemura, N. Vermeulen, J. Van Erps, D. A. B. Miller, and H. Thienpont, “Design of large scale plasmonic nanoslit arrays for arbitrary mode conversion and demultiplexing,” Opt. Express 22(1), 646–660 (2014). [CrossRef] [PubMed]
22. A. Rickman, “The commercialization of silicon photonics,” Nat. Photonics 8(8), 579–582 (2014). [CrossRef]
23. A. E. J. Lim, J. Song, F. Qing, C. Li, X. Tu, N. Duan, K. K. Chen, R. P. C. Tern, and T. Y. Liow, “Review of silicon photonics foundry efforts,” IEEE J. Sel. Top. Quantum Electron. 20(4), 405–416 (2014). [CrossRef]
24. R. Soref, “Silicon photonics: a review of recent literature,” Silicon 2(1), 1–6 (2010). [CrossRef]
25. L. Zhang, X. Tan, M. Yang, M. Qi, T. Hu, and J. Yang, “On-chip wavelength-routed photonic networks with comb switches,” in Proc. 9th IEEE Int. Conf. GFP, Aug. 2012, pp. 279–281. [CrossRef]
26. L. Lu, M. Zhang, F. Zhou, W. Chang, J. Tang, D. Li, X. Ren, Z. Pan, M. Cheng, and D. Liu, “Inverse-designed ultra-compact star-crossings based on PhC-like subwavelength structures for optical intercross connect,” Opt. Express 25(15), 18355–18364 (2017). [CrossRef] [PubMed]
27. J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antenn. Propag. 52(2), 397–407 (2004). [CrossRef]
28. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).
29. C. Sun, M. T. Wade, Y. Lee, J. S. Orcutt, L. Alloatti, M. S. Georgas, A. S. Waterman, J. M. Shainline, R. R. Avizienis, S. Lin, B. R. Moss, R. Kumar, F. Pavanello, A. H. Atabaki, H. M. Cook, A. J. Ou, J. C. Leu, Y. H. Chen, K. Asanović, R. J. Ram, M. A. Popović, and V. M. Stojanović, “Single-chip microprocessor that communicates directly using light,” Nature 528(7583), 534–538 (2015). [CrossRef] [PubMed]
30. D. Celo, D. J. Goodwill, P. Dumais, J. Jiang, and E. Bernier, “Low-loss waveguide crossings for photonic integrated circuits on SOI technology”, Proc. Group IV Photon. Conf., 189–190 (2014). [CrossRef]
31. S.-H. Kim, G. Cong, H. Kawashima, T. Hasama, and H. Ishikawa, “Tilted MMI crossings based on silicon wire waveguide,” Opt. Express 22(3), 2545–2552 (2014). [CrossRef] [PubMed]
32. Y. Li, C. Xu, C. Zeng, W. Wang, J. Yang, H. Yu, and X. Jiang, “Hybrid plasmonic waveguide crossing based on the multimode interference effect,” Opt. Commun. 335, 86–89 (2015). [CrossRef]
