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Abstract
We present the wafer-level characterization of a 256-channel optical phased array operating at 1550 nm, allowing the sequential testing of different OPA circuits without any packaging steps. Using this, we establish that due to random fabrication variations, nominally identical circuits must be individually calibrated. With this constraint in mind, we present methods that significantly reduce the time needed to calibrate each OPA circuit. In particular, we show that for an OPA of this scale, a genetic optimization algorithm is already >3x faster than a simple hill climbing algorithm. Furthermore, we describe how the phase modulators within the OPA may be individually characterized ‘in-situ’ and how this information can be used to configure the OPA to emit at any arbitrary angle following a single, initial calibration step.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Significant effort is currently being directed towards the cost reduction of spatial imaging systems to enable wide scale deployment, notably in the automotive sector. The replacement of assemblies of discrete optical components by a highly integrated solid-state solution would represent a significant step in this direction. The use of optical phased arrays (OPAs) to achieve rapid beam-steering is one significant area of study [1,2]. Since the initial demonstrations of optical phased arrays based on silicon photonics (SiPh) over ten years ago [3], the maturity of OPAs based on SiPh has steadily increased, but complete systems based on this beam scanning method are still in their infancy. With several notable exceptions [4–7], the majority of demonstrations have shown OPAs with less than 100 optical channels. However, in order to simultaneously meet system requirements in terms of beam power, field of view (FOV) and beam divergence, it is likely that OPAs with several hundred, if not thousands of optical channels are required [1,8]. The challenges of packaging and operating such large-scale circuits have meant that few have been demonstrated. In the following, we describe the wafer-level calibration of a relatively large scale 256 channel silicon-based optical phased array operating at a wavelength, λ = 1550 nm. In avoiding the necessity to package the chips for testing, we are able to compare nominally identical circuits on different dies and to devise strategies for rapid calibration.
2. Description of tested circuit and characterization equipment
2.1 256 channel optical phased array
Figure 1 shows images of the OPA circuit that was used for this study. The Nch = 256 OPAs were fabricated on the 200 mm pilot line at CEA-LETI [9]. The process is carried out on silicon on insulator (SOI) wafers with a 300 nm-thick Si layer, which is processed with several patterning steps, leading to a choice of waveguides thicknesses of 300, 280 or 150 nm.
A standard fiber grating coupler is used to couple light into the circuit (insertion loss (IL) of ∼2 dB at 1.55 µm). The light is then divided into 256 parallel waveguides via an 8-stage MMI tree (IL of each 1 × 2 MMI is ∼0.1 dB). The waveguide pitch in the phase modulator region is 5 µm. The phase of each optical channel is individually controlled using a thermal phase modulator (PM) in which a 10/110 nm Ti/TiN layer is placed and patterned into a linear 125 µm-long heating element 600 nm above the Si waveguide layer. Parallel air filled trenches are etched into the SiO2 either side of the heater element and the silicon waveguide. These act to improve the modulation efficiency from Pπ ∼ 20 mW (without trenches) to Pπ ∼ 6 mW, where Pπ is the required power to induce a π phase shift, and to virtually eliminate thermal crosstalk between neighboring waveguides, even for a waveguide pitch of 2 µm, as is reported in [8].
The output antennas (Fig. 1 top-right) are formed from using a single 20 nm-deep etch of an enlarged, 1.2 µm-wide waveguide. The grating period is 575 nm and the filling factor is 50%. The antenna pitch is d = 1.5 µm. Therefore, the theoretical minimum beam divergence for this OPA is Δφ3dB ∼ λ⁄Nchd = 4 mRad (∼0.2°) and the maximum unambiguous FOV is ± sin-1(λ⁄2d) = ±0.54 rad ∼ ±31°.
Since the thermal modulator pitch (5 µm) is larger than the antenna pitch, it is necessary to include a waveguide fan-in section. This section is not designed to equalize the optical path length between channels. Although the waveguide length in this section varies by only ∼120 µm between channels, giving a negligible insertion loss difference (waveguide propagation loss ∼1 dB/cm), the phase relations between antennas are random by design.
2.2 Prober based OPA characterization
Figure 2 shows the main elements of our prober based characterization system [10,11], which is based on a standard 300mm probing station (Cascade Elite 300) typically used in the microelectronic industry for circuit testing. Due to space constraints, the OPA emission is imaged directly on the bare sensor of an InGaAs camera (Allied Vision Goldeye G-033) with no intermediate lenses. This limits the angular imaging range to ∼ ±5° given the width of the camera sensor (∼ 1 cm) and the minimum distance to the wafer (∼ 5.5 cm). To allow wider angle characterization, the camera is mounted on a computer controlled linear actuator. The camera can be accurately displaced to allow the characterization of the OPA farfield emission in the range -30°<φ<30° with an angular resolution better than 0.02°/pixel. It should be noted that in this case, the camera-wafer distance is less than the Fraunhoffer distance, and is therefore not strictly in the farfield, however the approximation is small and does not affect the presented results.
Light is injected into the circuit via a standard single-mode fiber. A probe card was custom made to provide 256 electrical contacts to corresponding pads on the OPA circuit via two staggered rows of 128 pins while allowing access for the input optical fiber. The card is placed above the wafer, and once fixed in position, allows reliable and repeatable electrical contact to all the chip-side contact pads as the tested circuits are changed using the moveable prober chuck. The experiments were performed at room temperature and no active temperature control was applied to the wafer and/or the prober chuck.
All 256 electrical channels are individually controlled by a custom made driver circuit consisting of 512 12-bit DACs (of which half are used here) and current amplifiers allowing up to 6 V and 80 mA per channel.
3. Results
3.1 Single angle OPA calibration
When no voltage is applied to any of the PMs in the circuit, the light emitted by the OPA antennas is randomly distributed in the φ-plane (as defined is Fig. 6(a)). This is due to a random phase distribution at the antennas caused by both the unequal optical length of the OPA channels (due to the fan-in section previously described) and the unavoidable random variations in waveguide dimensions arising from imperfections in the fabrication process [12]. In order to obtain the linear phase distribution required to create an output beam in the farfield with the minimum possible divergence at the desired φ-angle, it is necessary to apply an appropriate set of command values to the PMs. This may be achieved by defining a suitable figure of merit (FOM) to describe the beam quality and applying a multi-parameter optimization algorithm. There are many available choices for the definition of the FOM, each with various advantages and disadvantages [13]. In this study, we choose to optimize the least squares fit between the one dimensional (xy-plane) far-field profile and the theoretical ideal OPA emission profile computed from the array factor equation [14].
The so called ‘hill-climbing’ algorithm is commonly used to solve this type of multi-parameter optimization [15,16]. In this application, the FOM is recorded while the power dissipated in a single PM is swept over values corresponding to phase shifts in the range 0 → 2π. The PM setting corresponding to the maximum FOM is retained and applied before performing the same operation on the following PM. The minimum number of FOM evaluations is therefore Nch * NDAC, where NDAC is the number of discrete phase modulation levels between 0 and 2π. Although our driver circuit allows up to 212 = 4096 discrete voltage levels, a much lower number proves sufficient [17]. Here we apply 6-bit (64 level) discretization, meaning that a single optimization iteration requires Nch * NDAC = 16384 FOM evaluations. The time taken to perform the hill climbing algorithm therefore increases linearly with Nch. The acquisition frequency of our camera is 50Hz, so in our case this calibration takes 5 - 6 minutes.
Among many interesting approaches to this multi-parameter optimization problem [18–21], so-called ‘genetic algorithms’ provide an alternative [22–25] to the commonly-used hill climbing algorithm. Rather than considering the effect of each PM sequentially on the FOM (or ‘fitness’), this type of algorithm begins by evaluating the FOM of sets of randomly generated values (genes), retaining successful sets (individuals) to form the basis of future ‘generations’ to converge towards an optimum solution. Unlike the hill climbing algorithm, the number of FOM evaluations required to converge towards an ideal solution is not proportional to Nch.
Figure 3(a) shows the FOM as a function of the number of FOM evaluations for ten consecutive applications of the hill-climbing (blue) as well as the genetic (red) algorithm on a single OPA. It can be seen that for our OPA with Nch = 256, the genetic algorithm converges with ∼5000 FOM evaluations, which is around a third of that is required for the hill climbing algorithm. This advantage will become increasingly important for future OPA as Nch increases into the hundreds or thousands [10]. It can also be seen in Fig. 3(b) that the normalized amplitude of main beam side lobes tends towards the theoretically minimum value of -13 dB (for a perfect sinc function) only when using the genetic algorithm. Indeed, while the solution to which the hill-climbing algorithm converges is influenced by the initial conditions (i.e. the phase relations in the uncalibrated circuit), initial randomization in the genetic algorithm increases the probability of converging to the absolute optimum solution.
Fig. 3. (a) FOM as a function of FOM evaluations for the hill climbing (blue) and genetic (red) calibration algorithms. (b) Output beam profiles resulting from consecutive calibrations using each algorithm.
3.2 Phase variations in nominally identical circuits
The possibility of wafer-level testing enables the consecutive characterization of nominally identical circuits. This allows us to evaluate to what extent the phase relations in OPA circuits of this scale are subject to random variations, and therefore to illustrate the necessity that each circuit must be individually calibrated. Figure 4 shows the far-field image obtained from nominally identical circuits from different dies of the same 200mm wafer using the same set of DAC values obtained from the calibration performed on the circuit from die number 5. It can be seen that while some low intensity peaks can be observed (contrary to the emission profile from the uncalibrated OPA shown in Fig. 4(b)), the well-formed spot from the initially calibrated circuit is not present on neighboring dies. This shows that the relative phase variations between waveguides in the parallel sections of the circuit (the waveguide length between the first power splitter and the antenna section is ∼2.5mm) are not reproducible. This confirms the necessity that each OPA must be calibrated separately.
Fig. 4. (a) Tested dies over the 200 mm wafer. (b) Output field of a non-calibrated OPA. (c) Farfield images of OPA output beam when the calibration obtained from a circuit on die 5 is applied to identical circuits on neighboring dies. All images use the same camera settings.
3.3 Wafer-level variations of calibrated OPA beams
Calibrating individually the nominally identical circuits over the wafer surface allows other aspects of the OPA beam quality to be studied. Figure 5(a) shows the beam divergence of individually calibrated OPAs over the wafer dies. The possibility of achieving the minimum theoretical beam divergence (Δφ3dB ∼ 4 mRad ∼0.2°) in the φ-plane is confirmed on all dies over the wafer. This is to be expected, given the possibility of actively optimizing the phase of each antenna. However, the beam properties in the θ-direction are determined by the characteristics of the antenna diffraction grating itself and cannot be actively optimized after fabrication. As such, Fig. 5(b) shows that the beam direction can vary over the surface of the wafer. This can be attributed to local variations in the etch depth and grating filling factor. While the actual antenna grating length was 1.7mm, these variations also affected the effective antenna length and therefore the divergence value in theta, whose median value was Δθ3dB ∼ 5 mRad ∼0.3°.
Fig. 5. Wafer-scale variations of the obtained output beam divergence in φ (a) and the beam direction in θ (b).
3.4 Single calibration OPA beam sweeping
It is possible to calibrate separately each OPA for every possible emission angle as in [26], creating a large lookup table to be consulted during the OPA operation, allowing the beam to be redirected as desired. The disadvantage of this approach would be a time consuming calibration process as well a driving circuit with sufficient memory and memory access rate to facilitate beam steering at the desired frequency [27]. However, in the following we show that this is not necessary.
Figure 6(a) shows a schematic representation of the phase in each waveguide of an uncalibrated OPA. Phase variations upstream of the PM section, ΔψprePM, may arise from a combination of designed and fabrication-induced differences in the optical path length. These variations may be greater than 2π, but the relevant value is ΔψprePM modulo 2π, which is represented by the red line. Additional phase variations are also present downstream of the PMs, ΔψpostPM represented in green (that may also include fixed fabrication-induced phase variations within the PMs themselves). For unactivated phase modulators, the sum of these phase variations, ΔψprePM + ΔψpostPM, will lead to a random phase profile at the antennas (blue line) and a random farfield emission profile in φ. The calibration of the OPA via the optimization algorithm serves to compensate the phase variations in each channel in order to achieve a linear phase front. Figure 6(b) represents a calibrated OPA (for φ = 0). The magenta line represents the phase modulation applied to each channel to compensate ΔψprePM + ΔψpostPM. Once this initial calibration has been performed using the optimization algorithm, the OPA may be configured to emit a beam in any direction by adding a constant phase step, $\Delta \psi = 2\pi d\sin (\mathrm{\varphi } )/\lambda$, across the antenna array. This linear phase gradient (dotted magenta line in Fig. 6(c)) must be added to the phase modulation required to compensate the phase variations. Since it is not practically possible, or desirable from a power consumption point of view, to apply phase shift values beyond 2π (in our OPA, even for $\mathrm{\varphi } = 1^\circ $, ${N_{ch}} \times \Delta \psi \gt 8\pi $), the addition of a linear phase gradient must be accompanied by modulo 2π operation, as depicted in Fig. 6(d).
Fig. 6. Relative phase relations between OPA channels. (a) Uncalibrated OPA (b), calibrated for φ = 0°, (c) application of a linear phase gradient for φ ≠ 0° and (d) application of modulo 2π function.
To use this method of configuring the OPA for an arbitrary emission angle therefore requires a means of evaluating the DAC value that corresponds to $\Delta \psi = $ 2π. This may be achieved by characterizing a separate test structure, such as a Mach Zehnder interferometer containing the same PM that is used in the OPA circuit. There is, however, a more direct way of measuring the phase modulator efficiency within the OPA circuit itself. Figure 7(a) shows the beam intensity (defined as the integral of the detected response in an area around the emission peak corresponding to the FWHM of the emitted beam) of a calibrated OPA as a function of the power dissipated in a single phase modulator. Despite the relatively small influence of a single optical channel among the 256 channels in the whole circuit, a sinusoidal dependence can be clearly observed and fitted to obtain the modulation efficiency (from the sinus function period) of the PM under test. Using this method, we can therefore individually characterize each phase modulator in the OPA. Evaluating the PM efficiency in this way is preferable to doing so in a separate test structure as it takes into account the specific electrical characteristics and the thermal environment of the PM in the circuit, which may differ from that of a standalone test structure.
Fig. 7. (a) In-situ evaluation of phase modulator behavior. (b) Variation of P2π within the 256ch OPA.
Figure 7(b) shows the efficiency of each PM of an OPA measured using this ‘in-situ’ method. The mean P2π value is 10.9 mW with a standard deviation of σ = ±0.66 mW. It should be noted that the accuracy of the obtained values depends on several parameters such as the quality of the sinus fit function, which is itself dependent on the noise in the data. In our case, we attribute the majority of this noise to temporal variations in fiber – chip coupling efficiency since the fiber is subject to vibrations as it is not fixed to the chip to allow wafer movement on the probing station. Moreover, actual variations in the Pπ values may arise from differences in electrical access resistance. In the tested OPA, the heating element resistance is ∼1.8 KΩ and the access resistance varies by several tens of ohms due to unequal electrical path lengths. Applying this method to OPAs with higher Nch may require the application of methods to increase the signal-to noise ratio, such as synchronous detection.
Using the data from Fig. 7 it would be possible to carry out the modulo 2π function to an applied linear gradient using the extracted P2π values of each individual modulator. However, in our case, we observe that the calibration is equivalent when using a single P2π value. Therefore, after having calibrated the OPA at φ = 0°, thus obtaining the power applied to the nth PM, Pn(φ = 0°), to obtain any φ, we apply a linear phase gradient, ΔP, and associated modulo 2π function, such that Pn(φ ≠ 0°) = [Pn(φ = 0°) + n.ΔP)] mod(α), where α is the PM power used for the modulo operation. Ideally, α = P2π, otherwise, discontinuities are introduced into the linear phase front, reducing the beam intensity. Figure 8 shows the measured beam intensity for φ = 1°, 5°, 10°, 15° and 20° as a function of α. The intensity values are normalized with respect to the beam intensity value obtained by performing a separate calibration (using the genetic algorithm) at the target angle. It can be seen that when an α value equal to the mean P2π value previously measured is used, i.e. α = | P2π |, the normalized beam power is equal to, or slightly greater than unity. This means that the calibration quality is equal (or marginally superior) to that obtained using the genetic algorithm for the target angle. The simulated dependence (for φ = 5°) using the array factor equation [14] (dashed black line in Fig. 8) matches closely the measured data. It can be noted that even for α → | P2π | /2, the beam intensity is still over half that for α = | P2π |. Also, since the dependence resembles a sin2 function, with the optimum α value at the center of the peak, the beam intensity is a slowly varying function of α, making this method usefully robust. It should be noted, that the applicability of this method relies, to some extent, on negligible phase modulator crosstalk, such as in our case, or for OPAs using electro-optic phase modulators.
Fig. 8. Beam intensity versus the PM power used for the modulo function, α, for target angles, φ = 1°, 5°, 10°, 15° and 20°. Curves are normalized to beam power obtained via a separate calibration at the target angle.
Figure 9(a) shows the superposed output beam profiles obtained using a single initial calibration (at φ = 0°) and applying linear phase gradients to achieve beam angles of -20° to 20° in steps equal to the beam divergence of 0.2°. This represents the minimum number of discrete beam directions required to fill the field of view considered (see close-up in Fig. 9(b)). Had these 201 beam angles been calibrated separately, this would have required 201 × 5000 = 106 FOM evaluations (∼5.5 hours at an acquisition rate of 50 Hz) and a lookup table requiring >5 × 104 DAC values (∼37 kB for 6-bit values). Using the calibration method detailed above, these time and memory requirements are significantly reduced to 5000 FOM evaluations (∼100s) and 256 6-bit DAC values. Figure 9(c) shows the extracted beam divergence values for each angle compared to the simulated values. The measured values are within 0.01 - 0.02° of the simulated values over the whole considered FOV. The offset between the simulated and the measured beam divergence can be attributed to a small error in the measurement bench angular calibration. The irregular beam power envelope is caused by a combination of suboptimal antenna grating directionality [28] and crosstalk between adjacent antennas.
Fig. 9. (a) Measured beam power for φ = -20 to 20° in steps of 0.2° using a single initial calibration. (b) Zoom showing the FOV coverage. (c) Extracted and simulated beam divergence versus beam angle.
4. Conclusion
We have described the prober-based characterization of a 256-channel silicon-based optical phased array operating at a wavelength of 1550 nm. The possibility of wafer-level measurements has allowed us to demonstrate clearly that separate calibrations must be performed on nominally identical circuits from different dies due to fabrication induced phase errors. With this requirement in mind, our results show that genetic algorithms offer a clear advantage with respect to simpler ‘hill-climbing’ type algorithms, which is already significant for the 256-channel OPA used here, but will become even greater as the OPA channel count increases further. Furthermore, for OPAs with negligible phase modulator crosstalk, such as in our case, we demonstrate that by performing an in-situ evaluation of the phase modulator efficiency, it is possible to configure the OPA to emit at any angle following a single initial calibration step, substantially reducing the time to calibrate each circuit for subsequent use.
We believe that these results contribute to the industrialization of integrated optical phased array technology.
Funding
ECSEL Vizta European project (826600); Programme d’investissement d’avenir, IRT Nanoelec (ANR-10-AIRT-05); Institut Carnot / ANR.
Acknowledgments
The authors would like to thank François Simoens, Jérôme Meilhan, Jean Hue, Vincent Moulin and Michael Scott for useful discussions.
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. C.-P. Hsu, B. Li, B. S. Rivas, A. 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]
2. X. Sun, L. Zhang, Q. Zhang, and W. Zhang, “Si Photonics for Practical LiDAR Solutions,” Appl. Sci. 9(20), 4225 (2019). [CrossRef]
3. K. Van Acoleyen, W. Bogaerts, J. Jágerská, N. Le Thomas, R. Houdré, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477 (2009). [CrossRef]
4. S. A. Miller, Y.-C. Chang, C. T. Phare, M. C. Shin, M. Zadka, S. P. Roberts, B. Stern, X. Ji, A. Mohanty, O. A. J. Gordillo, U. D. Dave, and M. Lipson, “Large-scale optical phased array using a low-power multi-pass silicon photonic platform,” Optica 7(1), 3–6 (2020). [CrossRef]
5. 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]
6. C. V. Poulton, M. J. Byrd, B. Moss, E. Timurdogan, R. Millman, and M. R. Watts, “8192-Element Optical Phased Array with 100° Steering Range and Flip-Chip CMOS,” in 2020 Conference on Lasers and Electro-Optics (CLEO) (2020), pp. 1–2.
7. Y. Li, B. Chen, Q. Na, Q. Xie, M. Tao, L. Zhang, Z. Zhi, Y. Li, X. Liu, X. Luo, G. Lo, F. Gao, X. Li, J. Song, and J. Song, “Wide-steering-angle high-resolution optical phased array,” Photonics Res. 9(12), 2511–2518 (2021). [CrossRef]
8. D. R. Fowler, S. Guerber, J. Faugier-Tovar, N. A. Tyler, and B. Szelag, “Integrated optical phased array based on a binary splitter tree with reduced number of control voltages,” J. Light. Technol. 40(12), 4027–4032 (2022). [CrossRef]
9. Q. Wilmart, H. El Dirani, N. Tyler, D. Fowler, S. Malhouitre, S. Garcia, M. Casale, S. Kerdiles, K. Hassan, C. Monat, X. Letartre, A. Kamel, M. Pu, K. Yvind, L. K. Oxenløwe, W. Rabaud, C. Sciancalepore, B. Szelag, and S. Olivier, “A Versatile Silicon-Silicon Nitride Photonics Platform for Enhanced Functionalities and Applications,” Appl. Sci. 9(2), 255 (2019). [CrossRef]
10. S. Guerber, D. Fowler, I. Charlet, P. Grosse, K. Abdoul-Carime, J. Faugier-Tovar, and B. Szelag, “Development, calibration and characterization of silicon photonics based optical phased arrays,” in Smart Photonic and Optoelectronic Integrated Circuits XXIII (SPIE11690, 22–29 (2021).
11. N. A. Tyler, D. Fowler, S. Malhouitre, S. Garcia, P. Grosse, W. Rabaud, and B. Szelag, “SiN integrated optical phased arrays for two-dimensional beam steering at a single near-infrared wavelength,” Opt. Express 27(4), 5851 (2019). [CrossRef]
12. Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Opt. Express 23(13), 16890 (2015). [CrossRef]
13. T. Komljenovic and P. Pintus, “On-chip calibration and control of optical phased arrays,” Opt. Express 26(3), 3199 (2018). [CrossRef]
14. R. L. Haupt, Antenna Arrays: A Computational Approach (John Wiley & Sons, 2010).
15. J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Opt. Express 23(5), 5861–5874 (2015). [CrossRef]
16. Q. Wang, Q. Wang, S. Wang, S. Wang, L. Jia, Y. Cai, W. Yue, and M. Yu, “Silicon nitride assisted 1 × 64 optical phased array based on a SOI platform,” Opt. Express 29(7), 10509–10517 (2021). [CrossRef]
17. J. Ø. Kjellman, T. D. Kongnyuy, B. Figeys, S. Kerman, M. Dahlem, X. Rottenberg, P. Soussan, and R. Jansen, “Minimizing DAC Complexity for Control of Optical Phased Arrays,” in 2019 IEEE Photonics Conference (IPC) (2019), pp. 1–2.
18. L. Leng, Z. Zeng, G. Wu, Z. Lin, X. Ji, Z. Shi, and W. Jiang, “Phase calibration for integrated optical phased arrays using artificial neural network with resolved phase ambiguity,” Photonics Res. 10(2), 347 (2022). [CrossRef]
19. J. Jin, E.-S. Lee, K.-W. Chun, S.-S. Lee, and M.-C. Oh, “Fast-running beamforming algorithm for optical phased array beam scanners comprised of polymeric waveguide devices,” Opt. Express 30(2), 768–779 (2022). [CrossRef]
20. L. Li, W. Chen, X. Zhao, and M. Sun, “Fast Optical Phased Array Calibration Technique for Random Phase Modulation LiDAR,” IEEE Photonics J. 11(1), 1–10 (2019). [CrossRef]
21. H. Zhang, Z. Zhang, C. Peng, and W. Hu, “Phase Calibration of On-Chip Optical Phased Arrays via Interference Technique,” IEEE Photonics J. 12(2), 1–10 (2020). [CrossRef]
22. Q. Liu, Y. Lu, B. Wu, P. Jiang, R. Cao, J. Feng, J. Guo, and L. Jin, “Silicon Optical Phased Array Side Lobe Suppression Based on an Improved Genetic Algorithm,” in Asia Communications and Photonics Conference/International Conference on Information Photonics and Optical Communications 2020 (ACP/IPOC) (2020), Paper T2D.3 (Optica Publishing Group, 2020), p. T2D.3.
23. J. He and L. Kang, “On the convergence rates of genetic algorithms,” Theor. Comput. Sci. 229(1-2), 23–39 (1999). [CrossRef]
24. M. C. Shin, A. Mohanty, K. Watson, G. R. Bhatt, C. T. Phare, S. A. Miller, M. Zadka, B. S. Lee, X. Ji, I. Datta, and M. Lipson, “Chip-scale blue light phased array,” Opt. Lett. 45(7), 1934–1937 (2020). [CrossRef]
25. C. T. Phare, M. C. Shin, S. A. Miller, B. Stern, and M. Lipson, “Silicon optical phased array with high-efficiency beam formation over 180 degree field of view,” ArXiv Prepr. ArXiv180204624 (2018).
26. L. Zhang, Y. Li, B. Chen, Y. Wang, H. Li, Y. Hou, M. Tao, Y. Li, Z. Zhi, X. Liu, X. Li, Q. Na, Q. Xie, M. Zhang, X. Li, F. Gao, X. Luo, G.-Q. Lo, and J. Song, “Two-dimensional multi-layered SiN-on-SOI optical phased array with wide-scanning and long-distance ranging,” Opt. Express 30(4), 5008 (2022). [CrossRef]
27. M. Tao, T. Peng, C. Ding, J. Guan, Y. Li, L. Zhang, B. Chen, L. Gao, X. Li, J. Song, and F. Gao, “A Large-Range Steering Optical Phased Array Chip and High-Speed Controlling System,” IEEE Trans. Instrum. Meas. 71, 1–12 (2022). [CrossRef]
28. M. Raval, C. V. Poulton, and M. R. Watts, “Unidirectional waveguide grating antennas with uniform emission for optical phased arrays,” Opt. Lett. 42(13), 2563–2566 (2017). [CrossRef]
