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Abstract
In this work, we use the inverse design method to design three-channel and four-channel dual-mode waveguide crossings with the design regions of 4.32 µm-wide regular hexagon and 6.68 µm-wide regular octagon, respectively. Based on the highly-symmetric structures, the fundamental transverse electric (TE0) and TE1 modes propagate through the waveguide crossings efficiently. Moreover, the devices are practically fabricated and experimentally characterized. The measured insertion losses and crosstalks of the three-channel and dual-mode waveguide crossing for both the TE0 and TE1 modes are less than 1.8 dB and lower than −18.4 dB from 1540 nm to 1560 nm, respectively. The measured insertion losses of the four-channel and dual-mode waveguide crossing for the TE0 and TE1 modes are less than 1.8 dB and 2.5 dB from 1540 nm to 1560 nm, respectively, and the measured crosstalks are lower than −17.0 dB. In principle, our proposed scheme can be extended to waveguide crossing with more channels and modes.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, the rapid improvement of the nanofabrication technologies has triggered the explosive trends toward the high-density integration in nanophotonics. Various nanophotonic devices, such as power splitters [1,2], mode (de)multiplexers [3,4], polarization beam splitters [5,6] and so on, have been demonstrated on the silicon-on-insulator (SOI) platform because of a high refractive index contrast which enables tight optical confinement with a typical waveguide dimension about hundreds of nanometers. With the fast growth of the device density in photonics integrated circuits (PICs), a significant increase in the number of optical waveguide crossings could be inevitable. In contrast to the electronics where multi-layer interconnects allow flexible routing, current PICs are constricted to a single layer. Thus, the high-density PICs are difficult or even impossible to be realized without low-loss and ultra-compact waveguide crossings.
In fact, a lot of efforts have been focused on the efficient waveguide crossings for a long time. Since lateral confinement in a direct waveguide crossing is lost near the crossing thereby causing the light to diffract, dramatically mitigating strong diffraction effects at the crossing region is the key of various design schemes. Different schemes for optimizing the performance of the waveguide crossing have been reported, including expanded crossing [7], multi-mode interference (MMI) [8], subwavelength grating [9], and Maxwell’s fisheye lens (MFL) [10]. Although these devices exhibit very attractive performance, they can hardly feature compact footprint, multiple channels, and multiple modes at the same time. For example, Hongnan Xu et al. realized an MMI-based 2 × 2 waveguide crossing with the insertion loss (IL) of less than 1.5 dB and crosstalk (CT) of lower than −18 dB for both the fundamental transverse magnetic (TM0) and TM1 modes, but the MMI length was about 28.16 µm [11]. Inverse design algorithms, such as direct-binary-search (DBS) algorithm [12,13], genetic algorithm [14,15], objective-first algorithm [16,17], and deep learning [18,19], can provide more degrees of freedom to explore the complex light behaviors in the digital structures at the sub-wavelength scale. Accordingly, several waveguide crossings have been designed based on the inverse design methods. Hailong Han et al. used the particle swarm optimization to design a 1 µm × 1 µm waveguide crossing with average IL of less than 0.2 dB and CT of lower than −37 dB [20]. This device occupies compact footprint and satisfactory performance, but it only supports the single mode. In order to support more modes, the dual-mode and three-mode 2 × 2 waveguide crossings with compact footprints were demonstrated [21,22]. For example, Yingjie Liu et al. employed the DBS algorithm to design a 2 × 2 three-mode crossing, which exhibited the footprint of 8 µm × 8 µm, ILs of less than 0.82 dB, and CTs of lower than -20 dB [22]. However, the 2 × 2 channels may limit the data capacity. The number of channels is an important feature for a waveguide crossing. Luluzi Lu et al. proposed several inverse-designed 4 × 4, 5 × 5, and 6 × 6 star crossings based on the photonic-crystal-like subwavelength structure, but the devices cannot support high-order modes [23]. It can be seen the ultra-compact, multi-channel, and multi-mode waveguide crossings for the high-density PICs are still under pursuit.
In this work, we propose the three-channel and four-channel dual-mode waveguide crossings. By the inverse design optimization, one can dramatically suppress the reflection and diffraction in the design region. In addition, based on the highly-symmetric structures, both the fundamental transverse electric (TE0) and TE1 modes can effectively propagate in each channel with great uniformity. In principle, this design method can be extended to waveguide crossings with more channels and modes. The proposed devices may have great impact and practicality for the high-density and multi-mode optical intercross system.
2. Designs and simulations
The DBS algorithm, one of the famous inverse design algorithms, has the advantages of simple principle, fast convergence, and fabrication robustness. The core concept of the DBS algorithm is that after digitalizing the design region into many pixels, the algorithm greatly controls the light behaviors by searching the pixel state one by one. The device design region is discretized into many pixels. The pixel size is generally about 100 nm × 100 nm, which means the DBS-based device are greatly robust to fabrication errors. As shown in Fig. 1, the state of the randomly chosen pixel is switched and then a figure of merit (FOM) is calculated. The FOM corresponds to the device performance. The pixel state is retained if the device performance goes up. Otherwise, this pixel returns to previous state and the algorithm proceeds to the next pixel. The DBS algorithm continues to search the next pixel until the device performance does not improve further.
We employ the DBS algorithm to optimize the waveguide crossings on the SOI platform with a 220 nm-thick top silicon layer and an air cladding. It is well known that the DBS algorithm is quite sensitive to the initial structure and tends to converge prematurely in local optima [24]. To achieve a better performance device, we manually set an initial structure rather than a random one to start the searching process of the DBS algorithm. The first challenge is to find a proper way to arrange air holes in the design region of the waveguide crossings. We choose the highly symmetric shape as the design region shape aiming to realize the same functionality in each waveguide. Besides, the next-row air holes are arranged in the gap between the previous-row adjacent ones. This arrangement allows the air holes to be more evenly distributed in the design region. We here choose regular hexagon as the design region shape of the three-channel and dual-mode waveguide crossing. Although a precise-calculation method was proposed to design highly-symmetric pixel distribution [23], we here adopt a simply geometric-symmetry method, detailed in our previous work [25], to realize the initial structure. As is shown in Fig. 2(a), the design region of regular hexagon is divided into twelve symmetric parts. After filling one of the parts with the air holes in Fig. 2(b), the the air-hole distributions of the other parts in Fig. 2(c) can be obtained by symmetrical geometric relations.
Fig. 2. Design process of the initial structure of the three-channel and dual-mode waveguide crossing. (a) Design region with twelve symmetric parts. (b) One of the twelve symmetric parts fully filled with the air holes. (c) Initial pattern.
A three-dimensional diagram of the three-channel and dual-mode waveguide crossing is illustrated in Fig. 3(a). The crossing consists of three 0.9 µm-wide waveguides and one design region of 4.32 µm-wide regular hexagon. The row spacing of the air holes is 104 nm and the column spacing is 120 nm. The diameter of the air hole is 90 nm, and the depth is 220 nm. We use the DBS algorithm to facilitate the device optimization. The finite-difference time domain, a method of approximately solving Maxwell’s equations, is used for the numerical simulation. In the optimization process, the air hole distributions still maintain symmetry, which can not only perfectly realize crossing functionality but significantly reduce time cost. The top view of the optimized structure is shown in Fig. 3(b). The simulated electric field distributions for the TE0 and TE1 modes are shown in Fig. 3(c), where the black lines are the device profiles. At the same time, the simulated ILs for the TE0 and TE1 modes in Fig. 3(d) are less than 0.5 dB and 0.7 dB from 1540 nm to 1560 nm, respectively, and the CTs are all lower than −60 dB.
Fig. 3. Design and simulated results of the three-channel and dual-mode waveguide crossing. (a) Three-dimensional diagram. (b) Top view. (c) Simulated electric field distributions. (d) Simulated transmission spectra.
To realize the four-channel and dual-mode waveguide crossing, we choose the regular octagon as the design region shape. As shown in Fig. 4(a), one can use the symmetrical geometric relations to get the initial structure with sixteen symmetric parts. The crossing consists of four 0.9 µm-wide waveguides and one design region of 6.68 µm-wide regular octagon. The row spacing of the air holes is 99.4 nm, and the column spacing is 160 nm. The diameter of the air hole is 90 nm, and the depth is 220 nm. After the DBS algorithm optimization, the optimized structure is obtained in Fig. 4(b). The simulated electric field distributions for the TE0 and TE1 modes are shown in Fig. 4(c). At the same time, the simulated ILs for the TE0 and TE1 modes in Fig. 4(d) are less than 0.7 dB and 1.4 dB from 1540 nm to 1560 nm, respectively, and the CTs are all lower than −55.7 dB.
Fig. 4. Design and simulated results of the four-channel and dual-mode waveguide crossing. (a) Three-dimensional diagram. (b) Top view. (c) Simulated electric field distributions. (d) Simulated transmission spectra.
To further investigate the fabrication tolerance of the proposed devices, we modified the diameters of etching holes from 80 nm to 100 nm and simulated the performance of the three-channel and four-channel dual-mode waveguide crossings with the previously optimized structures, respectively. As shown in Fig. 5(a), the simulated ILs of the three-channel and dual-mode waveguide crossing for the TE0 and TE1 modes are still less than 1.2 dB and 2.0 dB from 1540 nm to 1560 nm under diameter variations from -5 nm to 10 nm, respectively. The CTs are still lower than -57.8 dB. In addition, as shown in Fig. 5(b), the simulated ILs of the four-channel and dual-mode waveguide crossing for the TE0 and TE1 modes are still less than 1.3 dB and 2.5 dB from 1540 nm to 1560 nm under diameter variations from -5 nm to 5 nm, respectively. The CTs are still lower than -49.7 dB. Therefore, one may estimate the rate of inductively coupled plasma etching before etching the device and control appropriate etching errors by setting an appropriate etching time.
Fig. 5. Fabrication tolerances of the (a) three-channel and dual-mode waveguide crossing and (b) four-channel and dual-mode waveguide crossing.
3. Fabrications and measurements
The designed structures are fabricated and demonstrated experimentally. The devices are patterned on SOI substrate with 220 nm top silicon and 2 µm-thick buried oxide by the e-beam lithography system. An inductively coupled plasma etcher is used to transfer the pattern to the silicon device layer. The devices are characterized by the set-up consisting of a broadband optical source (ASE-CL-100-T-B), a vertical fiber-chip coupling stage, an optical spectrum analyzer (Yokogawa AQ6370D), and a power meter. The light polarization is controlled by a polarization controller before entering the grating coupler of the tested device. The transmission spectra are acquired by testing the output waveguide of the devices via an optical spectrum analyzer.
The scanning electron microscope (SEM) image of the fabricated three-channel and dual-mode waveguide crossing is shown in Fig. 6(a), and the inset (red dashed box) describes that an extra reference circuit with a back-to-back mode (de)MUX (MDM system) is fabricated to normalize the transmission and to extract the loss of our devices. The detailed SEM images of the fabricated mode (de)MUX and three-channel and dual-mode waveguide crossing are shown in Fig. 6(b) and Fig. 6(c), respectively. The performance of the reference MDM system is characterized in Fig. 6(d), in which, for example, “I1 – O1” denotes the transmission spectrum from input port 1 (I1) to output port 1 (O1). The TE0 and TE1 modes are excited when the light is launched in input port I1 and I2, respectively. As shown, the measured ILs for both the TE0 and TE1 modes are less than 2.7 dB from 1540 nm to 1560 nm. As a result, we can estimate the IL of a single mode (de)MUX to be less than 1.4 dB. The CTs are lower than –17.2 dB from 1540 nm to 1560 nm. In addition, the transmission spectrum of the entire device consisting of the proposed crossing and mode (de)MUXs are characterized in Fig. 6(e). As shown, the measured ILs for both the TE0 and TE1 modes are less than 4.5 dB, which means that the ILs of our fabricated crossing should be less than 1.8 dB from 1540 nm to 1560 nm. The measured CTs for both modes are lower than −18.4 dB.
Fig. 6. Fabrication and experimental results of the fabricated three-channel and dual-mode waveguide crossing. (a) SEM image of the fabricated device consisting of one crossing and two mode (de)MUXs. Insets: an extra reference MDM circuit. (b) Detailed SEM image of the mode (de)MUX. (c) Detailed SEM image of the three-channel and dual-mode waveguide crossing. (d) Normalized measured transmission spectra of the fabricated reference MDM circuit. (e) Normalized measured transmission spectra of entire fabricated device consisting of one crossing and two mode (de)MUXs.
Similarly, we use the same measurement method to verify the performance and functionality of the four-channel and dual-mode waveguide crossing. The SEM image of the entire fabricated device consisting of one crossing and two mode (de)MUXs are shown in Fig. 7(a). At the same time, the detailed SEM images of the fabricated mode (de)MUX and four-channel and dual-mode waveguide crossing are shown in Fig. 7(b) and Fig. 7(c), respectively. In addition, when the light is launched from the I1 and I2 ports, the normalized transmission spectra measured in the O1 and O2 ports are plotted in Fig. 7(d). As shown, the measured ILs for the TE0 and TE1 modes are less than 4.5 dB and 5.2 dB, which means that the ILs of our fabricated crossing for the TE0 and TE1 modes should be less than 1.8 dB and 2.5 dB from 1540 nm to 1560 nm. The measured CTs for both modes are lower than −17.0 dB.
Fig. 7. Fabrication and experimental results of the four-channel and dual-mode waveguide crossing. (a) SEM image of the fabricated device consisting of one crossing and two mode (de)MUXs. (b) Detailed SEM image of the mode (de)MUX. (c) Detailed SEM image of the four-channel and dual-mode waveguide crossing. (d) Normalized measured transmission spectra of entire fabricated device consisting of one crossing and two mode (de)MUXs.
4. Discussions
4.1 Initial structural analysis
The DBS algorithm is quite sensitive to the initial structure. We optimize the three-channel and dual-mode waveguide crossing with the different initial structures in the DBS algorithm. Figure 8(a), Fig. 8(b), Fig. 8(c), and Fig. 8(d) show the silicon-filled, air-filled, random, and preset initial structures and corresponding finally optimized structures, respectively. Figure 8(e) shows the FOM convergences in the optimization processes. The FOM is defined as: $\textrm{FOM = }T{E_0} + T{E_1}$, where TE0 and TE1 stand for the different-mode transmittances in the output waveguide of the three-channel and dual-mode waveguide crossing. One can find that the initial FOM and final performance are higher, when the preset structure is chosen as the initial structure (blue curve). It suggests that the high-quality initial structure contributes to the optimization of the DBS algorithm.
Fig. 8. Optimization processes with different initial structures. (a) Initial structure filled with the silicon pixels. (b) Initial structure filled with the air holes. (c) Initial structure filled with the randomly-distributed air holes. (d) Initial structure filled with the preset-distributed air holes. (e) FOM converges as a function of the iterations.
4.2 Performance comparison
We take the three-channel direct waveguide crossing as an example to compare with the proposed one. Figure 9(a) shows the three-channel direct waveguide crossing consisting of three 0.9 µm-wide waveguides. The included angle between each waveguide is 60 degrees. The simulated electric field distributions for the TE0 and TE1 modes are shown in Fig. 9(b). As shown, when the TE0 and TE1 modes propagate through the central crossing region, they diverge significantly because of the light diffraction. The ILs for the TE0 and TE1 modes in Fig. 9(c) are less than 2.6 dB and 11.1 dB, respectively. Compared with the direct waveguide crossing, the proposed DBS-based crossings dramatically suppress the diffraction in the design region, and the TE0 and TE1 modes efficiently propagate from the input ports to the output ports.
Fig. 9. Design and simulated results of the three-channel direct waveguide crossing. (a) Top view. (b) Simulated electric field distributions. (c) Simulated transmission spectra.
In order to further embody the specific strengths of our proposed device, we summarize and compare the waveguide crossings designed by different schemes in Table 1. In row 1 and row 2, the waveguide crossings designed by the traditional methods exhibited the satisfactory performances, multiple channels, and two modes, but they had the large footprints and cannot support the TE-polarization modes. In row 3 and row 4, the waveguide crossings designed by the inverse design methods had not only compact footprints but also multiple channels, but they only supported the TE0 mode. Compared the devices in row 3 and row 4, the devices in row 5 and row 6 increased the number of the TE-polarization modes, but they only had 2 × 2 channels. On the whole, the waveguide crossings with multiple channels and multiple TE-polarization modes are still absent. Our proposed devices occupy three and four channels and support the TE0 and TE1 modes, which can greatly build the high-density and multi-mode PICs. More importantly, the scheme we proposed also can be extended to waveguide crossing with more channels and modes.
Table 1. Performance comparison of different waveguide crossings
5. Conclusion
Based on the highly symmetric structures, we use the inverse design method to design three-channel and four-channel dual-mode waveguide crossings. The three-channel and dual-mode waveguide crossing consists of three 0.9 µm-wide waveguides and one design region of 4.32 µm-wide regular hexagon. The four-channel and dual-mode waveguide crossing consists of three 0.9 µm-wide waveguides and one design region of 6.68 µm-wide regular octagon. We simulate the performances and investigate the fabrication tolerances. Moreover, the devices are practically fabricated and experimentally characterized. The measured ILs and CTs of the three-channel and dual-mode waveguide crossing for both the TE0 and TE1 modes are less than 1.8 dB and lower than −18.4 dB from 1540 nm to 1560 nm, respectively. The measured ILs of the four-channel and dual-mode waveguide crossing for the TE0 and TE1 modes are less than 1.8 dB and 2.5 dB from 1540 nm to 1560 nm, respectively, and the measured CTs are lower than −17.0 dB. In principle, our proposed scheme can be extended to waveguide crossing with more channels and modes. The proposed devices may offer an attractive approach to promote practical applications for on-chip optical interconnections.
Funding
Program for New Century Excellent Talents in University (NCET-12-0142); Natural Science Foundation of Hunan Province (13JJ3001); Foundation of NUDT (JC13-02-13, ZK17-03-01); China Postdoctoral Science Foundation (2018M633704); National Key Research and Development Program of China (2022YFF0706005); National Natural Science Foundation of China (12272407, 60907003, 61805278, 62275269, 62275271).
Acknowledgments
Hansi Ma contributed to the idea, performed the numerical simulation, collected the data, and wrote the original draft. Hansi Ma and Junbo Yang fabricated the devices. Hansi Ma and Xin He measured the devices. Junbo Yang, Yuanxi Peng, and Liang Fang directed the project. All authors discussed the results and revised the manuscript.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. Ma, J. Huang, K. Zhang, and J. Yang, “Ultra-compact and efficient 1 × 2 mode converters based on rotatable direct-binary-search algorithm,” Opt. Express 28(11), 17010–17019 (2020). [CrossRef]
2. K. Xu, L. Liu, X. Wen, W. Sun, N. Zhang, N. Yi, S. Sun, S. Xiao, and Q. Song, “Integrated photonic power divider with arbitrary power ratios,” Opt. Lett. 42(4), 855–858 (2017). [CrossRef]
3. H. Xie, Y. Liu, S. Wang, Y. Wang, Y. Yao, Q. Song, J. Du, Z. He, and K. Xu, “Highly compact and efficient four-mode multiplexer based on pixelated waveguides,” IEEE Photonics Technol. Lett. 32(3), 166–169 (2020). [CrossRef]
4. D. Dai, C. Li, S. Wang, H. Wu, Y. Shi, Z. Wu, S. Gao, T. Dai, H. Yu, and H. Tsang, “10-channel mode (de) multiplexer with dual polarizations,” Laser Photonics Rev. 12(1), 1700109 (2018). [CrossRef]
5. D. Dai and H. Wu, “Realization of a compact polarization splitter-rotator on silicon,” Opt. Lett. 41(10), 2346–2349 (2016). [CrossRef]
6. Y. Liu, S. Wang, Y. Wang, W. Liu, H. Xie, Y. Yao, Q. Song, X. Zhang, Y. Yu, and K. Xu, “Subwavelength polarization splitter–rotator with ultra-compact footprint,” Opt. Lett. 44(18), 4495–4498 (2019). [CrossRef]
7. 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]
8. Y. Zhang, A. Hosseini, X. Xu, D. Kwong, and R. T. Chen, “Ultralow-loss silicon waveguide crossing using Bloch modes in index-engineered cascaded multimode-interference couplers,” Opt. Lett. 38(18), 3608–3611 (2013). [CrossRef]
9. P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, D. Xu, S. Janz, A. Densmore, and T. J. Hall, “Subwavelength grating crossings for silicon wire waveguides,” Opt. Express 18(15), 16146–16155 (2010). [CrossRef]
10. H. Xu and Y. Shi, “Metamaterial-based maxwell’s fisheye lens for multimode waveguide crossing,” Laser Photonics Rev. 12(10), 1800094 (2018). [CrossRef]
11. H. Xu and Y. Shi, “Dual-mode waveguide crossing utilizing taper-assisted multimode-interference couplers,” Opt. Lett. 41(22), 5381–5384 (2016). [CrossRef]
12. 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]
13. H. Ma, M. Luo, J. He, T. Du, Z. Zhang, X. Jiang, X. He, L. Fang, and J. Yang, “Inverse design of nonvolatile reconfigurable mode generator and optical circulator based on a novel concept of a fully-digitized module,” J. Lightwave Technol. 40(24), 7869–7878 (2022). [CrossRef]
14. X. Jiang, H. Yuan, D. Chen, Z. Zhang, T. Du, H. Ma, and J. Yang, “Metasurface based on inverse design for maximizing solar spectral absorption,” Adv. Opt. Mater. 9(19), 2100575 (2021). [CrossRef]
15. C. Lu, Z. Liu, Y. Wu, Z. Xiao, D. Yu, H. Zhang, C. Wang, X. Hu, Y. Liu, X. Liu, and X. Zhang, “Nanophotonic polarization routers based on an intelligent algorithm,” Adv. Opt. Mater. 8(10), 1902018 (2020). [CrossRef]
16. J. Huang, J. Yang, D. Chen, W. Ba, J. Han, Z. Zhang, J. Zhang, X. He, Y. Han, and L. Liang, “Implementation of on-chip multi-channel focusing wavelength demultiplexer with regularized digital metamaterials,” Nanophotonics 9(1), 159–166 (2020). [CrossRef]
17. J. Huang, J. Yang, D. Chen, X. He, Y. Han, J. Zhang, and Z. Zhang, “Ultra-compact broadband polarization beam splitter with strong expansibility,” Photonics Res. 6(6), 574–578 (2018). [CrossRef]
18. H. Yuan, Z. Wang, Z. Peng, J. Wu, and J. Yang, “Ultra-compact and nonvolatile nanophotonic neural networks,” Adv. Opt. Mater. 2300215 (2023).
19. C. Wu, H. Yu, S. Lee, R. Peng, I. Takeuchi, and M. Li, “Programmable phase-change metasurfaces on waveguides for multimode photonic convolutional neural network,” Nat. Commun. 12(1), 96 (2021). [CrossRef]
20. H. Han, H. Li, X. Zhang, A. Liu, T. Lin, Z. Chen, H. Lv, M. Lu, X. Liu, and Y. Chen, “High performance ultra-compact SOI waveguide crossing,” Opt. Express 26(20), 25602–25610 (2018). [CrossRef]
21. W. Chang, L. Lu, X. Ren, D. Li, Z. Pan, M. Cheng, D. Liu, and M. Zhang, “Ultracompact dual-mode waveguide crossing based on subwavelength multimode-interference couplers,” Photonics Res. 6(7), 660–665 (2018). [CrossRef]
22. Y. Liu, K. Xu, S. Wang, W. Shen, H. Xie, Y. Wang, S. Xiao, Y. Yao, J. Du, Z. He, and Q. Song, “Arbitrarily routed mode-division multiplexed photonic circuits for dense integration,” Nat. Commun. 10(1), 3263 (2019). [CrossRef]
23. 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]
24. H. Ma, J. Huang, K. Zhang, and J. Yang, “Arbitrary-direction, multichannel and ultra-compact power splitters by inverse design method,” Opt. Commun. 462(1), 125329 (2020). [CrossRef]
25. H. Ma, J. Huang, K. Zhang, and J. Yang, “Inverse-designed arbitrary-input and ultra-compact 1×N power splitters based on high symmetric structure,” Sci. Rep. 10(1), 11757 (2020). [CrossRef]
26. 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]
