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Abstract
Non-uniformly distributed gratings on the silicon-on-insulator platform for one-dimensional beam steering are designed by direct binary search inverse-design method. The gratings exhibit good emission directionality and far-field characteristics. Within a relatively small wavelength tuning range of 1517-1577 nm, the longitudinal scanning angle for TE and TM light is 23.65° and 10.81°, respectively, both of which are much larger than their uniform counterparts. By polarization multiplexing and etching depth optimization, a remarkable longitudinal scanning angle of 32.10° and high beam steering efficiency of 0.55°/nm are obtained. This work may pave the way for the development of miniaturized optical phased arrays with excellent beam steering performance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Radars play a significant role in military affairs and for civil use. Traditional microwave and millimeter wave radars have been widely used after long-term development. With the increasing demand of gathering information from the environment, the detection precision of radars needs to be improved. Light Detection and Ranging (Lidar) sensors have attracted considerable attention in recent years, which have great application potential in navigation [1], mapping [2], driverless car [3], intelligent robot [4] and other emerging applications because of their advantages of obtaining distance and speed accurately, high resolution, and short wavelength. Up to date, different types of lidars have been developed, including mechanical lidars and solid-state lidars based on MEMS, flash, and optical phased array (OPA). Compared with other technologies, OPAs have a lot of advantages, such as small size, controllable direction and fast scanning speed, which are consistent with lidar’s future development trend [5]. Particularly, considering the significant advantages in fabrication process, integration and cost, silicon-based integrated optical waveguide OPAs have become the most potential solution for miniaturized lidars.
Passive scanning structures which use wavelength modulation to realize beam steering become a development trend [6–8]. Traditionally, beam steering can be realized by thermal optical phase tuning. When it comes to scan in both longitudinal and lateral directions, the N × N beam scanner array consists of N2 thermal phase tuners, which significantly increase the structural complexity and power consumption. Wavelength modulation is another way to realize beam steering, particularly in longitudinal direction. To date, beam steering by wavelength modulation based on uniform gratings has been widely reported [9,10]. However, as the angle dispersion ability of uniform gratings is relatively weak, the tuning efficiency is severely limited. Studies have shown that the limited tuning efficiency of uniform gratings puts forward higher requirements for the tuning ability of lasers to produce a wide wavelength range (usually more than 100 nm). According to grating equation, if the period of uniform gratings is larger than a half wavelength, the maximum scanning range will be severely reduced [11]. Non-uniformly distributed grating structures are expected to be a feasible way to expand the scanning ability in the longitudinal direction. The multiplexing of gratings’ periods has been proven to improve the beam steering range [12]. However, traditional design methods are not capable of finding the optimal non-uniformly distributed grating structure in an infinite structural parameter space. In recent years, inverse design has gained increasing attention due to its potential in the design of ultra-compact high-performance optical devices. This approach combines optimization with electromagnetic calculation, which is able to find optimal structure in infinite parameter space to reach the performance goal. Up to date, a great deal of optical devices, such as polarization beam splitters [13], optical logical gates [14], optical varifocal mentalens [15], and optical waveguide crossings [16], have been demonstrated by inverse design method. However, inverse-designed gratings for beam steering have not been reported yet.
On the other hand, polarization is also a factor limiting the ability of wavelength modulation. The grating emitters based on silicon-on-insulator (SOI) waveguides are usually polarization sensitive [17]. The scanning ranges of TE and TM modes are not completely coincident due to the difference in the refractive indexes. Thus, polarization multiplexing is expected to be an efficient approach of enhancing the beam steering ability.
In this work, direct binary search (DBS) algorithm is adopted to create a new kind of grating outcoupler on SOI platform. The inverse-designed non-uniformly distributed gratings couple light into air and realize beam steering by wavelength tuning. Results show that compared with uniform gratings, non-uniformly distributed grating can enlarge the scanning range in the longitudinal direction. The scanning ability is improved even further by applying polarization multiplexing. Non-uniformly distributed gratings designed with both TE and TM modes have a large scanning angle of 23.95° in the longitudinal direction from 1517 nm to 1577 nm and the beam steering efficiency is improved to 0.41°/nm. After optimizing the etching depth, the overall scanning range reaches 32.10° in the longitudinal direction, and its corresponding beam steering efficiency is 0.55°/nm from 1517 nm to 1577 nm.
2. Design and simulation
The mainstream inverse design methods include adjoint method, objective first method and direct binary research (DBS) algorithm, etc. Adjoint method uses only two full-field simulations to get the gradient for all design degree of freedoms, but has problems with discrete objective function. Objective first method has the advantages of fast speed and combining global optimal process with local optimal process [18]. However, the device performance after binarization may greatly degrade compared with the initial device structure, and high fabrication process is required. Compared with other inverse design methods, DBS algorithm has the advantages of fast convergence speed, easy fabrication, and the ability to deal with discrete objective function, which has been widely used in the design of ultra-compact optical devices with various functions. Hence in this work, DBS is used to design non-uniform gratings. DBS method is a traversing algorithm which was firstly used for designing nano-photonic structure by Bing Shen, etc. [19]. The process of DBS algorithm is illustrated in Fig. 1. DBS algorithm starts with a randomly-generated initial device pattern, which is defined by a randomly-generated binary matrix. With this specific initial binary matrix, the figure-of-merit (FOM) of the corresponding initial device pattern can be calculated. Then the logical state of each pixel starts to be reversed one at a time. That is, the pixel state can be switched from “0” to “1”, or the “1” state of a pixel can be changed into “0” state. If the change of pixel’s state results in an improved FOM, the new pixel state will be reserved. Otherwise the pixel goes back to the original state. One iteration finishes after each and every pixel is reversed in the above-mentioned way. After one iteration ends, the next iteration begins, and the optimized pattern of last iteration is adopted as the new initial device pattern. All the pixels are reversed one by one and corresponding FOMs are inspected once again. The optimization doesn’t stop until FOM equals that of the last iteration [13]. In order to analyze the performance of designed gratings, the FOM of DBS algorithm is defined as follows,
where θ represents the scanning angle of gratings’ main lobe with the maximum peak value at a certain wavelength at far field. As is shown in Fig. 2, θmax and θmin are the maximum and minimum longitudinal scanning angles of non-uniformly distributed gratings in the wavelength range of 1517-1577 nm, respectively.Figure 3(a) and (b) show the designed non-uniformly distributed grating structures for TE and TM modes, respectively. Each structure consists of eight gratings which perform as slits in multi-slit interference apparatus [12]. The non-uniformly distributed gratings are patterned on an SOI substrate. The silicon layer is 290 nm thick with a permittivity of εSi = 11.97 (refractive index 3.46) [13], which is placed on top of a silica buffer layer with a permittivity of εSilica = 2.10 (refractive index 1.45) [20]. The distance between two adjacent waveguides is set to be 1.80 μm. The gratings are 450 nm wide and there are 60 pixels on each grating. Pixels are rectangular with the size of 450 nm × 500 nm × 70 nm. Since the overall size of non-uniformly distributed grating structure is determined by pixel size and the number of pixels, the total length of designed non-uniformly distributed gratings is 30 μm. The pixels’ material is indicated by two states “0” and “1”: “1” for silicon and “0” for air. For air pixels, the etching depth is set to be 70 nm. In order to explain the corresponding relationship between logic states and pixel’s material, some indicators are plotted below the cross-sections of gratings in Fig. 3. It can be seen that when the logic state of a pixel is set to “0”, the corresponding position on the waveguide is shallow-etched and an air pixel is produced. While when the logic state of a pixel is “1”, the waveguide is not etched and a silicon swelling is left on the waveguide.
Fig. 3. Schematic diagrams of non-uniformly distributed gratings designed for (a) TE and (b) TM mode.
Non-uniformly distributed gratings can be regarded as a combination of uniform gratings with different periods, and the physical mechanism of non-uniformly distributed gratings is as follows. The beam steering angle of an OPA is adjusted by controlling the phase and/or amplitude of emitters in the array. The electro-magnetic field close to the emitters, i.e. the near field, can be fully controlled. The far field is described by Fraunhofer diffraction theory and is basically the complex Fourier transform of the near field [21]. Every pixel on the non-uniformly distributed gratings performs as an emitter. By changing the positions of these pixels, the phases of emitters are adjusted and then the near field of gratings can be controlled. By controlling the phase and/or amplitude of emitters, the near field can be fully controlled and the beam steering character will be adjusted.
3D-FDTD method is used to study the performance of designed structures [22]. Formulas of 3D-FDTD method are directly derived from MaxWell’s equations with the help of 3-D Yee’s mesh. Perfect math layers (PMLs) are also imposed at the edges of the computational window. At the far ends of input waveguides, eigenmode sources are set as the light sources to excite fundamental TE and TM modes. Through 3D-FDTD method, the far-field patterns of beam steering can be obtained, which can be used to assess the performance of the device. TE and TM fundamental modes are emitted vertically in the air by the designed non-uniformed distributed gratings, then the light beams can be detected in the far field and scanning angles can be measured. DBS algorithm is applied to optimize the performance of non-uniformly distributed gratings. The FOM in this work is defined as the difference between the maximum and minimum longitudinal scanning angles in the wavelength range of 1517-1577 nm. DBS algorithm starts with a random initial state of pixels, and the initial FOM can be calculated. Then the material of a randomly-chosen pixel will be changed. If the change of pixel’s material results in an improved FOM, the new pixel state will be reserved. Otherwise the pixel goes back to its original state. One iteration stops after each and every pixel is inspected in the above-mentioned way. The optimal pattern of the first iteration is set as the initial state of the next iteration. Then the algorithm proceeds and iterates. When the FOM stops to improve compared with the FOM of the last iteration, the program stops running. The “iteration vs FOM” curve is plotted in Fig. 4. By using a server with a 14-core central processing unit (Intel Xeon E5-2690), the optimization program iterated 6 times (number of iterations × pixel number = 6 × 60), and the total simulation time is about 65 hours for TE structure. For TM structure, optimization program iterated 7 times (number of iterations × pixel number = 7 × 60) and the simulation time is about 75 hours.
3. Results and discussion
3.1 Emission characteristics
2D simulation is used for estimating the emission directionality of non-uniformly distributed gratings. The simulation areas for TE mode and TM mode are plotted in Fig. 5(a) and Fig. 5(b), respectively. The simulation areas are surrounded by PML. SiO2 substrates are placed at the bottom of simulation areas and non-uniformly distributed gratings are on the top of substrates. Monitors are placed to obtain the upward transmission Ttop, downward transmission Tbottom, and backward transmission Tback reflected back to the input waveguides. The ratio (directionality = Ttop∕ (Tbottom + Ttop + Tback)) is used to characterize the emission directionality of the antenna [23]. Figure 5(d) shows the emission directionality of the two modes. It can be noticed that the directionality of TM mode fluctuates slightly between 20% and 27%, which is lower than that of TE mode. The directionality of TE mode varies from 55% to 88%, and the peak appears near the wavelength of 1550 nm. Researches show that the directionality of non-uniformly distributed gratings is almost the same as that of uniform gratings, so the non-uniformly distributed grating structures do not lead to the deterioration of directionality. Additionally, compared to the TM gratings, the TE gratings can better suppress the downward and backward radiation. Figure 5(c) shows the efficiency (coupled out energy / input port energy) of non-uniformly distributed gratings. According to the definition of directionality, directionality curve will be similar to the efficiency curve.
Fig. 5. Simulation structure of emission directionality for (a) TE and (b) TM mode. (c) and (d) Efficiency and directionality of non-uniformly distributed gratings.
3.2 Far field characteristics
The normalized far field patterns of non-uniformly distributed gratings are shown in Fig. 6. θ represents the angle in the longitudinal direction, which is modulated mainly by wavelength. Φ is the scanning angle in the lateral direction, which is adjusted by the structure of grating array. Figure 6(a) refers to the far field of TE polarized mode. For the designed 8-wire optical grating array, the full width at half maximum (FWHM) of TE mode beam is 3.54° in θ direction and 9.15° in Φ direction. Figure 6(b) shows normalization of intensity for TE mode with wavelengths of 1517, 1550 and 1577 nm, generating scanning angles of 41.0°, 49.6° and 54.5°, respectively. The FWHM of TE mode has good performance in the longitudinal direction. The beam width in longitudinal direction mainly depends on the number of pixels on one uniform grating. The more pixels, the smaller the FWHM. By increasing the number of waveguides and pixels, the beam width can be reduced and the resolution of beam steering will be improved. However, increasing the number of pixels will make the grating longer and affect other characteristics. In lateral direction, both the distance between two adjacent waveguides and the total number of waveguides have an effect on beam width. Figure 6(c) corresponds to the far field pattern of TM polarized mode, whose FWHM is 7.98° in θ direction and 7.96° in Φ direction. The scanning angles are 31.4°, 38.7° and 43.2°, respectively, as shown in Fig. 6(d). It can be noticed that the scanning beam of TE polarized mode is not affected by sidelobes, while there are significant sidelobes when scanning with TM polarized mode. By adding a term of sidelobe optimization to FOM, disturbance caused by grating lobes during scanning could be reduced in future work.
Fig. 6. Far-field distribution of non-uniformly distributed gratings at 1550 nm for (a) TE and (c) TM mode. (b) Normalization of intensity at φ=0 for TE mode with wavelengths of 1517, 1550 and 1577 nm. (d) Normalization of intensity at φ=0 for TM mode with wavelengths of 1517, 1550 and 1577 nm.
3.3 Beam steering characteristics
The longitudinal scanning angles of non-uniformly distributed gratings can be derived from their far-field distributions. The wavelength-dependent longitudinal scanning angles of non-uniformly distributed gratings are illustrated in Fig. 7. In Fig. 7(b), as the wavelength switches from 1517 nm to 1577 nm, the light beam for TE polarized mode steers from 40.31° to 57.84°, resulting in a scanning range of 17.53°. By comparison, the uniform gratings with the same feature size (the period is 1000 nm and the duty cycle is 0.5) only scan 5.04° in the same wavelength range. For TM polarized mode, non-uniformly distributed gratings steer from 33.89° to 44.70°, resulting in a longitudinal scanning angle of 10.81°. The scanning angle of its uniform counterpart is 6.81° in the same wavelength range. The results in Fig. 7 show that compared with uniform gratings, non-uniformly distributed grating structure can improve the performance of wavelength modulation in the longitudinal direction for both TE and TM polarized modes.
Fig. 7. Longitudinal scanning angles of uniform and non-uniform gratings for (a) TE and (b) TM mode, respectively.
Using polarization multiplexing is expected to expand the scanning angle in the longitudinal direction even further. It turns out that the scanning angles of gratings for TE and TM polarized mode do not completely coincide. In fact, they just overlap in a certain range. The non-uniformly distributed grating structures for TE and TM polarized modes are designed respectively, and the longitudinal scanning angles for both TE and TM modes are illustrated in Fig. 8(a). From 1517 to 1577 nm, the non-uniformly distributed gratings for TE polarized mode scan from 40.31° to 57.84°, and for TM polarized mode the beam steers from 33.89° to 44.70°. As is shown in Fig. 8(b), the overall θ angle of the designed polarization multiplexing structure reaches 23.95° in total, and its corresponding beam steering efficiency is 0.40°/nm within the 60 nm wavelength range from 1517 nm to 1577 nm. The results show that the superposition of scanning angles of TE and TM polarized modes can expand the scanning range, and the total scanning angle of TE and TM polarized mode is much larger than the scanning range of one single mode.
It is still possible to improve some aspects of the non-uniformly distributed gratings designed in this work. There are other considered ways to improve the performance in the next step. It can be seen in Fig. 7 that compared with traditional uniform grating, the scanning linearity of non-uniformly distributed gratings in θ direction is slacken because a compromise is made to keep a balance between large scanning angle and strong linearity. It is expected to find new approaches to optimize scanning linearity while the scanning range is expanded even further.
3.4 Etching characteristics
Researches show that when the etching depth of gratings change, the beam steering angles will also vary slightly. The scanning angles of non-uniformly distributed gratings with different etching depth are calculated. Previously the etching depth of the gratings is set to 70 nm, therefore, the scanning angles of gratings with different etching depth are compared with angles of gratings with 70 nm etching depth. The differences between scanning angles are plotted in Fig. 9 and (h) represent etching depth. Figure 9(a) and Fig. 9(b) show the differences of non-uniformly distributed gratings’ scanning angles with variant etching depth for TE and TM mode, respectively. As the etching depth increases from 50 nm to 80 nm, the scanning angles of both gratings shifts slightly from positive to negative along x-axis. It can be seen that the maximum variation of angles is about ± 9 ° for gratings designed for TE mode. While the scanning angles of non-uniformly distributed gratings for TM mode vary in a range of ±3°. It can be concluded that the etching depth has a notable influence on beam steering angles of TE gratings, but not on TM gratings.
Fig. 9. Dependence of beam steering angles on the etching depth gratings for (a) TE and (b) TM structures.
According to the results shown in Fig. 9, by adjusting the etching depth of the gratings designed for TE mode, the scanning range of the gratings can be fine-tuned more efficiently. The etching depth of the gratings for TE polarized modes is trimmed to h = 50 nm. Since trimming the etching depth has little effect on TM mode’s scanning range, the etching depth of TM grating is kept at h = 70 nm. After changing the etching depth, the longitudinal scanning angles for both TE and TM modes are illustrated in Fig. 10(a). After adjustment, the non-uniformly distributed gratings for TE polarized mode scan from 42.33° to 65.98° from 1517 to 1577 nm. For TM polarized mode the beam steering angles remain in the range of 33.89° to 44.70°. As is shown in Fig. 10(b), the overall θ angle consisting of both TE and TM longitudinal scanning ranges reaches 32.10° in total, and its corresponding beam steering efficiency is 0.55°/nm within the 58 nm wavelength range. The results show that changing the etching depth of the gratings can help fine-tune the beam steering range. The overlapping area of scanning ranges can be reduced, and the scanning range of the whole polarization multiplication gratings can be further expanded after adjusting the etching depth of gratings. Compared with those previously reported results, the longitudinal scanning angle and beam steering efficiency of wavelength-modulation OPA are improved, as shown in Table 1.
Fig. 10. Longitudinal scanning angles of non-uniformly distributed gratings when the etching depth of non-uniformly distributed gratings for TE mode is set to 50 nm.
Table 1. Summary and Comparisons with Previous Beam-Steering Structures
4. Conclusion
The utilization of inverse design for non-uniformly distributed gratings have been demonstrated for both TE and TM modes. The gratings exhibit good emission directionality and far field characteristics. In a relatively small wavelength tuning range of 1517-1577 nm, the longitudinal scanning angle for TE and TM light is 23.65°and 10.81°, respectively, both of which are much larger than the uniform gratings. By polarization multiplexing and etching depth optimization, a remarkable longitudinal scanning angle of 32.10° and high beam steering efficiency of 0.55°/nm are obtained. This work demonstrates that inverse design is a promising method for the realization of high performance optical phased arrays.
Funding
National Natural Science Foundation of China (61935003); State Key Laboratory of Information Photonics and Optical Communications (IPOC2020ZZ01, IPOC2022ZT02, IPOC2022ZZ01).
Disclosures
The authors declare no conflicts 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. N. Vandapel, D. F. Huber, A. Kapuria, and M. Hebert, “Natural terrain classification using 3-d lidar data,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE2004), pp. 5117–5122.
2. P. Potapov, X. Li, A. Hernandez-Serna, A. Tyukavina, M. C. Hansen, A. Kommareddy, A. Pickens, S. Turubanova, H. Tang, C. E. Silva, J. Armston, R. Dubayah, J. B. Blair, and M. Hofton, “Mapping global forest canopy height through integration of GEDI and Landsat data,” Remote Sens. Environ. 253, 112165 (2021). [CrossRef]
3. C.-P. Hsu, B. Li, B. Solano-Rivas, A. R. Gohil, P. H. Chan, A. D. Moore, and V. Donzella, “A review and perspective on optical phased array for automotive LiDAR,” IEEE J. Sel. Top. Quantum Electron. 27(1), 1–16 (2021). [CrossRef]
4. I. Kim, R. J. Martins, J. Jang, T. Badloe, S. Khadir, H.-Y. Jung, H. Kim, J. Kim, P. Genevet, and J. Rho, “Nanophotonics for light detection and ranging technology,” Nat. Nanotechnol. 16(5), 508–524 (2021). [CrossRef]
5. J. Chen and Y. Shi, “Research progress in solid-state LiDAR,” Opto-Electro. Eng. 46, 190218 (2019). [CrossRef]
6. K. V. Acoleyen, H. Rogier, and R. Baets, “Two-dimensional optical phased array antenna on silicon-on-insulator,” Opt. Express 18(13), 13655–13660 (2010). [CrossRef]
7. C. V. Poulton, A. Yaacobi, D. B. Cole, M. J. Byrd, M. Raval, D. Vermeulen, and M. R. Watts, “Coherent solid-state LIDAR with silicon photonic optical phased arrays,” Opt. Lett. 42(20), 4091–4094 (2017). [CrossRef]
8. S. Chung, H. Abediasl, and H. Hashemi, “A monolithically integrated large-scale optical phased array in silicon-on-insulator CMOS,” IEEE J. Solid-State Circuits 53(1), 275–296 (2018). [CrossRef]
9. T. Chan, E. Myslivets, and J. E. Ford, “2-Dimensional beamsteering using dispersive deflectors and wavelength tuning,” Opt. Express 16(19), 14617–14628 (2008). [CrossRef]
10. K. V. Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technol. Lett. 23(17), 1270–1272 (2011). [CrossRef]
11. M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50(12), 1732–1742 (2002). [CrossRef]
12. L.-X. Zhang, Y.-Z. Li, M. Tao, Y.-B. Wang, Y. Hou, B.-S. Chen, Y.-X. Li, L. Qin, F.-L. Gao, X.-S. Luo, G.-Q. Lo, and J.-F. Song, “Large-scale integrated multi-lines optical phased array chip,” IEEE Photonics J. 12(4), 1–8 (2020). [CrossRef]
13. W. Chang, X. Ren, Y. Ao, L. Lu, M. Cheng, L. Deng, D. Liu, and M. Zhang, “Inverse design and demonstration of an ultracompact broadband dual-mode 3 dB power splitter,” Opt. Express 26(18), 24135–24144 (2018). [CrossRef]
14. Q. Lu, X. Yan, W. Wei, X. Zhang, M. Zhang, J. Zheng, B. Li, Y. luo, Q. Lin, and X. Ren, “High-speed ultra-compact all-optical NOT and AND logic gates designed by a multi-objective particle swarm optimized method,” Opt. Laser Technol. 116, 322–327 (2019). [CrossRef]
15. M. Meem, S. Banerji, C. Pies, T. Oberbiermann, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Large-area, high-numerical-aperture multi-level diffractive lens via inverse design,” Optica 7(3), 252–253 (2020). [CrossRef]
16. 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]
17. X. Yan, J. Chen, D. Dai, and Y. Shi, “Polarization multiplexing silicon-photonic optical phased array for 2D wide-angle optical beam steering,” IEEE Photonics J. 13(2), 1–6 (2021). [CrossRef]
18. 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,” Nature Photon. 9(6), 374–377 (2015). [CrossRef]
19. 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]
20. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]
21. M. J. R. Heck, “Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering,” Nanophotonics 6(1), 93–107 (2017). [CrossRef]
22. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]
23. P. Ma, G. Luo, P. Wang, J. Ma, R. Wang, Z. Yang, X. Zhou, Y. Zhang, and J. Pan, “Unidirectional SiN antenna based on dual-layer gratings for LiDAR with optical phased array,” Opt. Commun. 501, 127361 (2021). [CrossRef]
24. P. F. Wang, G. Z. Luo, H. Y. Yu, Y. J. Li, M. Q. Wang, X. L. Zhou, W. X. Chen, Y. J. Zhang, and J. Q. Pan, “Improving the performance of optical antenna for optical phased arrays through high-contrast grating structure on SOI substrate,” Opt. Express 27(3), 2703–2712 (2019). [CrossRef]
25. W. Bogaerts, S. Dwivedi, R. Jansen, X. Rottenberg, and M. S. Dahlem, “A 2D pixelated optical beam scanner controlled by the laser wavelength,” IEEE J. Sel. Top. Quantum Electron. 27(1), 1–12 (2021). [CrossRef]
26. J. Chen, J. Wang, J. Li, Y. Yao, Y. Sun, J. Tian, Y. Zou, X. Zhao, and X. Xu, “Subwavelength structure enabled ultra-long waveguide grating antenna,” Opt. Express 29(10), 15133–15144 (2021). [CrossRef]
27. G. Zhou, S.-W. Qu, J. Wu, and S. Yang, “Design of a low-crosstalk sub-wavelength-pitch silicon waveguide array for optical phased array,” IEEE Photonics J. 13(4), 1–8 (2021). [CrossRef]
28. Y. Zhang, Y.-C. Ling, K. Zhang, C. Gentry, D. Sadighi, G. Whaley, J. Colosimo, P. Suni, and S. J. B. Yoo, “Sub-wavelength-pitch silicon-photonic optical phased array for large field-of-regard coherent optical beam steering,” Opt. Express 27(3), 1929–1940 (2019). [CrossRef]
